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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 Stack
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 Queue
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 Double-ended queue
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 Summary
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_hashing/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
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<span class="md-ellipsis">
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Chapter 6. Hash table
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</a>
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<label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
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Chapter 6. Hash table
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1 Hash table
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 Hash collision
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 Hash algorithm
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4 Summary
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</a>
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
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<div class="md-nav__link md-nav__container">
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<a href="../" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
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<span class="md-ellipsis">
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Chapter 7. Tree
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</span>
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</a>
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<label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
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Chapter 7. Tree
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1 Binary tree
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../binary_tree_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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7.2 Binary tree traversal
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../array_representation_of_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.3 Array Representation of tree
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</span>
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</a>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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<span class="md-ellipsis">
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7.4 Binary Search tree
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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<span class="md-ellipsis">
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7.4 Binary Search tree
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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Table of contents
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#741-operations-on-a-binary-search-tree" class="md-nav__link">
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<span class="md-ellipsis">
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7.4.1 Operations on a binary search tree
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</span>
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</a>
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<nav class="md-nav" aria-label="7.4.1 Operations on a binary search tree">
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#1-searching-for-a-node" class="md-nav__link">
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<span class="md-ellipsis">
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1. Searching for a node
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#2-inserting-a-node" class="md-nav__link">
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<span class="md-ellipsis">
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2. Inserting a node
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#3-removing-a-node" class="md-nav__link">
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<span class="md-ellipsis">
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3. Removing a node
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#4-in-order-traversal-is-ordered" class="md-nav__link">
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<span class="md-ellipsis">
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4. In-order traversal is ordered
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="#742-efficiency-of-binary-search-trees" class="md-nav__link">
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<span class="md-ellipsis">
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7.4.2 Efficiency of binary search trees
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#743-common-applications-of-binary-search-trees" class="md-nav__link">
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<span class="md-ellipsis">
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7.4.3 Common applications of binary search trees
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item">
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<a href="../avl_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.5 AVL tree *
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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7.6 Summary
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</a>
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</li>
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<span class="md-ellipsis">
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Chapter 8. Heap
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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Chapter 8. Heap
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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8.1 Heap
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</span>
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</a>
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8.2 Building a heap
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</span>
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</a>
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8.3 Top-k problem
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<span class="md-ellipsis">
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8.4 Summary
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</a>
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<a href="../../chapter_graph/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
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<span class="md-ellipsis">
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Chapter 9. Graph
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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Chapter 9. Graph
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1 Graph
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</span>
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</a>
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2 Basic graph operations
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 Graph traversal
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<a href="../../chapter_graph/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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9.4 Summary
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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Chapter 10. Searching
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</span>
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</a>
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<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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Chapter 10. Searching
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
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10.1 Binary search
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 Binary search insertion
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3 Binary search boundaries
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 Hashing optimization strategies
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</span>
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 Search algorithms revisited
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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10.6 Summary
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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<span class="md-ellipsis">
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Chapter 11. Sorting
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</span>
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</a>
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<label class="md-nav__title" for="__nav_13">
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<span class="md-nav__icon md-icon"></span>
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Chapter 11. Sorting
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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11.1 Sorting algorithms
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</span>
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</a>
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</li>
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.2 Selection sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.3 Bubble sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.4 Insertion sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.5 Quick sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.6 Merge sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.7 Heap sort
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.8 Bucket sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.9 Counting sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.10 Radix sort
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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11.11 Summary
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
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<span class="md-ellipsis">
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Chapter 12. Divide and conquer
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</span>
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</a>
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<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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Chapter 12. Divide and conquer
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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12.1 Divide and conquer algorithms
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</span>
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</a>
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
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12.2 Divide and conquer search strategy
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.3 Building binary tree problem
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 Tower of Hanoi Problem
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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12.5 Summary
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</span>
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</a>
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</li>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_backtracking/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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<span class="md-ellipsis">
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Chapter 13. Backtracking
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</span>
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</a>
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<label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_15">
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<span class="md-nav__icon md-icon"></span>
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Chapter 13. Backtracking
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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13.1 Backtracking algorithms
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
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13.2 Permutation problem
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<span class="md-ellipsis">
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13.3 Subset sum problem
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13.4 n queens problem
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13.5 Summary
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14.2 Characteristics of DP problems
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14.3 DP problem-solving approach¶
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14.4 0-1 Knapsack problem
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14.5 Unbounded knapsack problem
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14.6 Edit distance problem
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Chapter 15. Greedy
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15.1 Greedy algorithms
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</a>
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<h1 id="74-binary-search-tree">7.4 Binary search tree<a class="headerlink" href="#74-binary-search-tree" title="Permanent link">¶</a></h1>
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<p>As shown in Figure 7-16, a <u>binary search tree</u> satisfies the following conditions.</p>
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<ol>
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<li>For the root node, the value of all nodes in the left subtree <span class="arithmatex">\(<\)</span> the value of the root node <span class="arithmatex">\(<\)</span> the value of all nodes in the right subtree.</li>
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<li>The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition <code>1.</code> as well.</li>
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<p><a class="glightbox" href="../binary_search_tree.assets/binary_search_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Binary search tree" class="animation-figure" src="../binary_search_tree.assets/binary_search_tree.png" /></a></p>
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<p align="center"> Figure 7-16 Binary search tree </p>
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<h2 id="741-operations-on-a-binary-search-tree">7.4.1 Operations on a binary search tree<a class="headerlink" href="#741-operations-on-a-binary-search-tree" title="Permanent link">¶</a></h2>
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<p>We encapsulate the binary search tree as a class <code>BinarySearchTree</code> and declare a member variable <code>root</code> pointing to the tree's root node.</p>
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<h3 id="1-searching-for-a-node">1. Searching for a node<a class="headerlink" href="#1-searching-for-a-node" title="Permanent link">¶</a></h3>
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<p>Given a target node value <code>num</code>, one can search according to the properties of the binary search tree. As shown in Figure 7-17, we declare a node <code>cur</code>, start from the binary tree's root node <code>root</code>, and loop to compare the size between the node value <code>cur.val</code> and <code>num</code>.</p>
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<ul>
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<li>If <code>cur.val < num</code>, it means the target node is in <code>cur</code>'s right subtree, thus execute <code>cur = cur.right</code>.</li>
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<li>If <code>cur.val > num</code>, it means the target node is in <code>cur</code>'s left subtree, thus execute <code>cur = cur.left</code>.</li>
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<li>If <code>cur.val = num</code>, it means the target node is found, exit the loop, and return the node.</li>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:4"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1"><1></label><label for="__tabbed_1_2"><2></label><label for="__tabbed_1_3"><3></label><label for="__tabbed_1_4"><4></label></div>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example of searching for a node in a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_search_step1.png" /></a></p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_search_step2" class="animation-figure" src="../binary_search_tree.assets/bst_search_step2.png" /></a></p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_search_step3" class="animation-figure" src="../binary_search_tree.assets/bst_search_step3.png" /></a></p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_search_step4" class="animation-figure" src="../binary_search_tree.assets/bst_search_step4.png" /></a></p>
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<p align="center"> Figure 7-17 Example of searching for a node in a binary search tree </p>
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<p>The search operation in a binary search tree works on the same principle as the binary search algorithm, eliminating half of the cases in each round. The number of loops is at most the height of the binary tree. When the binary tree is balanced, it uses <span class="arithmatex">\(O(\log n)\)</span> time. The example code is as follows:</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:14"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label><label for="__tabbed_2_14">Zig</label></div>
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<div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">search</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-></span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">"""Search node"""</span>
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="n">cur</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span>
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<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="c1"># Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="c1"># Target node is in cur's right subtree</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o"><</span> <span class="n">num</span><span class="p">:</span>
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<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
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<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="c1"># Target node is in cur's left subtree</span>
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<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="k">elif</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">></span> <span class="n">num</span><span class="p">:</span>
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<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
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<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="c1"># Found target node, break loop</span>
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<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="k">break</span>
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<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="k">return</span> <span class="n">cur</span>
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</code></pre></div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Search node */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
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<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="c1">// Target node is in cur's right subtree</span>
|
|
|
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="c1">// Target node is in cur's left subtree</span>
|
|
|
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
|
|
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="p">;</span>
|
|
|
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="c1">// Found target node, break loop</span>
|
|
|
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="k">else</span>
|
|
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<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
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<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="c1">// Return target node</span>
|
|
|
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
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|
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="p">}</span>
|
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</code></pre></div>
|
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Search node */</span>
|
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
|
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<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
|
|
|
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w"> </span><span class="c1">// Target node is in cur's right subtree</span>
|
|
|
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w"> </span><span class="c1">// Target node is in cur's left subtree</span>
|
|
|
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
|
|
|
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w"> </span><span class="c1">// Found target node, break loop</span>
|
|
|
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w"> </span><span class="k">else</span>
|
|
|
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
|
|
|
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w"> </span><span class="c1">// Return target node</span>
|
|
|
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
|
|
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="p">}</span>
|
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Search</span><span class="p">}</span>
|
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{</span><span class="nx">binarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">search</span><span class="p">}</span>
|
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">search</span><span class="p">}</span>
|
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="no">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.zig</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">search</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<h3 id="2-inserting-a-node">2. Inserting a node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">¶</a></h3>
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<p>Given an element <code>num</code> to be inserted, to maintain the property of the binary search tree "left subtree < root node < right subtree," the insertion operation proceeds as shown in Figure 7-18.</p>
|
|
|
<ol>
|
|
|
<li><strong>Finding insertion position</strong>: Similar to the search operation, start from the root node, loop downwards according to the size relationship between the current node value and <code>num</code>, until the leaf node is passed (traversed to <code>None</code>), then exit the loop.</li>
|
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<li><strong>Insert the node at this position</strong>: Initialize the node <code>num</code> and place it where <code>None</code> was.</li>
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</ol>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_insert.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Inserting a node into a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_insert.png" /></a></p>
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<p align="center"> Figure 7-18 Inserting a node into a binary search tree </p>
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|
<p>In the code implementation, note the following two points.</p>
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<ul>
|
|
|
<li>The binary search tree does not allow duplicate nodes to exist; otherwise, its definition would be violated. Therefore, if the node to be inserted already exists in the tree, the insertion is not performed, and the node returns directly.</li>
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|
|
<li>To perform the insertion operation, we need to use the node <code>pre</code> to save the node from the previous loop. This way, when traversing to <code>None</code>, we can get its parent node, thus completing the node insertion operation.</li>
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</ul>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="k">def</span> <span class="nf">insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
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<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="w"> </span><span class="sd">"""Insert node"""</span>
|
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<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a> <span class="c1"># If tree is empty, initialize root node</span>
|
|
|
<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
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<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
|
|
|
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a> <span class="k">return</span>
|
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|
<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a> <span class="c1"># Loop find, break after passing leaf nodes</span>
|
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<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a> <span class="n">cur</span><span class="p">,</span> <span class="n">pre</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="kc">None</span>
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<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a> <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
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|
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a> <span class="c1"># Found duplicate node, thus return</span>
|
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<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="n">num</span><span class="p">:</span>
|
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<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a> <span class="k">return</span>
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<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a> <span class="n">pre</span> <span class="o">=</span> <span class="n">cur</span>
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<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a> <span class="c1"># Insertion position is in cur's right subtree</span>
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<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o"><</span> <span class="n">num</span><span class="p">:</span>
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<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
|
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<a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a> <span class="c1"># Insertion position is in cur's left subtree</span>
|
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<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
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<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a> <span class="c1"># Insert node</span>
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<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
|
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<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a> <span class="k">if</span> <span class="n">pre</span><span class="o">.</span><span class="n">val</span> <span class="o"><</span> <span class="n">num</span><span class="p">:</span>
|
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<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a> <span class="n">pre</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">node</span>
|
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<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a> <span class="k">else</span><span class="p">:</span>
|
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<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a> <span class="n">pre</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">node</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Insert node */</span>
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<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w"> </span><span class="c1">// If tree is empty, initialize root node</span>
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<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
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<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">;</span>
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<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w"> </span><span class="c1">// Found duplicate node, thus return</span>
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<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
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<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
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<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
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<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w"> </span><span class="c1">// Insertion position is in cur's right subtree</span>
|
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<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="p">;</span>
|
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|
<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w"> </span><span class="c1">// Insertion position is in cur's left subtree</span>
|
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<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w"> </span><span class="k">else</span>
|
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<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="p">;</span>
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<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w"> </span><span class="p">}</span>
|
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|
<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w"> </span><span class="c1">// Insert node</span>
|
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<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
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<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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|
<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w"> </span><span class="n">pre</span><span class="o">-></span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
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<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w"> </span><span class="k">else</span>
|
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<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w"> </span><span class="n">pre</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
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|
<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Insert node */</span>
|
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<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w"> </span><span class="c1">// If tree is empty, initialize root node</span>
|
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<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
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<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
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<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
|
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|
<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
|
|
|
<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w"> </span><span class="c1">// Found duplicate node, thus return</span>
|
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<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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|
<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
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|
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
|
|
|
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w"> </span><span class="c1">// Insertion position is in cur's right subtree</span>
|
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|
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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|
<a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w"> </span><span class="c1">// Insertion position is in cur's left subtree</span>
|
|
|
<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w"> </span><span class="k">else</span>
|
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|
<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
|
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<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w"> </span><span class="p">}</span>
|
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|
<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w"> </span><span class="c1">// Insert node</span>
|
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<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
|
|
|
<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
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|
<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
|
|
<a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w"> </span><span class="k">else</span>
|
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|
<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
|
|
|
<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="p">}</span>
|
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|
</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Insert</span><span class="p">}</span>
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|
</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{</span><span class="nx">binarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="bp">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="no">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.zig</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">insert</span><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>Similar to searching for a node, inserting a node uses <span class="arithmatex">\(O(\log n)\)</span> time.</p>
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<h3 id="3-removing-a-node">3. Removing a node<a class="headerlink" href="#3-removing-a-node" title="Permanent link">¶</a></h3>
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<p>First, find the target node in the binary tree, then remove it. Similar to inserting a node, we need to ensure that after the removal operation is completed, the property of the binary search tree "left subtree < root node < right subtree" is still satisfied. Therefore, based on the number of child nodes of the target node, we divide it into three cases: 0, 1, and 2, and perform the corresponding node removal operations.</p>
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<p>As shown in Figure 7-19, when the degree of the node to be removed is <span class="arithmatex">\(0\)</span>, it means the node is a leaf node and can be directly removed.</p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Removing a node in a binary search tree (degree 0)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case1.png" /></a></p>
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<p align="center"> Figure 7-19 Removing a node in a binary search tree (degree 0) </p>
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<p>As shown in Figure 7-20, when the degree of the node to be removed is <span class="arithmatex">\(1\)</span>, replacing the node to be removed with its child node is sufficient.</p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Removing a node in a binary search tree (degree 1)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case2.png" /></a></p>
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<p align="center"> Figure 7-20 Removing a node in a binary search tree (degree 1) </p>
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<p>When the degree of the node to be removed is <span class="arithmatex">\(2\)</span>, we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree "left subtree <span class="arithmatex">\(<\)</span> root node <span class="arithmatex">\(<\)</span> right subtree," <strong>this node can be either the smallest node of the right subtree or the largest node of the left subtree</strong>.</p>
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<p>Assuming we choose the smallest node of the right subtree (the next node in in-order traversal), then the removal operation proceeds as shown in Figure 7-21.</p>
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<ol>
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<li>Find the next node in the "in-order traversal sequence" of the node to be removed, denoted as <code>tmp</code>.</li>
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<li>Replace the value of the node to be removed with <code>tmp</code>'s value, and recursively remove the node <code>tmp</code> in the tree.</li>
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</ol>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:4"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1"><1></label><label for="__tabbed_4_2"><2></label><label for="__tabbed_4_3"><3></label><label for="__tabbed_4_4"><4></label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case3_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Removing a node in a binary search tree (degree 2)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step1.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case3_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_remove_case3_step2" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step2.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case3_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_remove_case3_step3" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step3.png" /></a></p>
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</div>
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<div class="tabbed-block">
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case3_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_remove_case3_step4" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step4.png" /></a></p>
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</div>
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</div>
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</div>
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<p align="center"> Figure 7-21 Removing a node in a binary search tree (degree 2) </p>
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<p>The operation of removing a node also uses <span class="arithmatex">\(O(\log n)\)</span> time, where finding the node to be removed requires <span class="arithmatex">\(O(\log n)\)</span> time, and obtaining the in-order traversal successor node requires <span class="arithmatex">\(O(\log n)\)</span> time. Example code is as follows:</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="5:14"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label><label for="__tabbed_5_14">Zig</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="k">def</span> <span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
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<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="w"> </span><span class="sd">"""Remove node"""</span>
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<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a> <span class="c1"># If tree is empty, return</span>
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<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a> <span class="k">return</span>
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<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a> <span class="c1"># Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a> <span class="n">cur</span><span class="p">,</span> <span class="n">pre</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="kc">None</span>
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<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a> <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a> <span class="c1"># Found node to be removed, break loop</span>
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<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="n">num</span><span class="p">:</span>
|
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<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a> <span class="k">break</span>
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<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a> <span class="n">pre</span> <span class="o">=</span> <span class="n">cur</span>
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<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a> <span class="c1"># Node to be removed is in cur's right subtree</span>
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<a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o"><</span> <span class="n">num</span><span class="p">:</span>
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<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
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<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a> <span class="c1"># Node to be removed is in cur's left subtree</span>
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<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a> <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
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<a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a> <span class="c1"># If no node to be removed, return</span>
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<a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a> <span class="k">if</span> <span class="n">cur</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
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<a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a> <span class="k">return</span>
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<a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a>
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<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a> <span class="c1"># Number of child nodes = 0 or 1</span>
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<a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a> <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a> <span class="c1"># When the number of child nodes = 0/1, child = null/that child node</span>
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<a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a> <span class="n">child</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span> <span class="ow">or</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
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<a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a> <span class="c1"># Remove node cur</span>
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<a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a> <span class="k">if</span> <span class="n">cur</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">:</span>
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<a id="__codelineno-28-29" name="__codelineno-28-29" href="#__codelineno-28-29"></a> <span class="k">if</span> <span class="n">pre</span><span class="o">.</span><span class="n">left</span> <span class="o">==</span> <span class="n">cur</span><span class="p">:</span>
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<a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a> <span class="n">pre</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">child</span>
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<a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a> <span class="n">pre</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">child</span>
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<a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a> <span class="k">else</span><span class="p">:</span>
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<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a> <span class="c1"># If the removed node is the root, reassign the root</span>
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<a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="n">child</span>
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<a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a> <span class="c1"># Number of child nodes = 2</span>
|
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<a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a> <span class="k">else</span><span class="p">:</span>
|
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<a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a> <span class="c1"># Get the next node in in-order traversal of cur</span>
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<a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a> <span class="n">tmp</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
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<a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a> <span class="k">while</span> <span class="n">tmp</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
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<a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a> <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">left</span>
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<a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a> <span class="c1"># Recursively remove node tmp</span>
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<a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a> <span class="bp">self</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
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<a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a> <span class="c1"># Replace cur with tmp</span>
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<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">val</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Remove node */</span>
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<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w"> </span><span class="c1">// If tree is empty, return</span>
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<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span>
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<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">;</span>
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<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w"> </span><span class="c1">// Found node to be removed, break loop</span>
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<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
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<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
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<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
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<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w"> </span><span class="c1">// Node to be removed is in cur's right subtree</span>
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<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
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<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="p">;</span>
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<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w"> </span><span class="c1">// Node to be removed is in cur's left subtree</span>
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<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w"> </span><span class="k">else</span>
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<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="p">;</span>
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<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w"> </span><span class="c1">// If no node to be removed, return</span>
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<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span>
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<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w"> </span><span class="c1">// Number of child nodes = 0 or 1</span>
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<a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w"> </span><span class="c1">// When the number of child nodes = 0 / 1, child = nullptr / that child node</span>
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<a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">nullptr</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="p">;</span>
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<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w"> </span><span class="c1">// Remove node cur</span>
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<a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
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<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w"> </span><span class="n">pre</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
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<a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w"> </span><span class="k">else</span>
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<a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w"> </span><span class="n">pre</span><span class="o">-></span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
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<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w"> </span><span class="c1">// If the removed node is the root, reassign the root</span>
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<a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
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<a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w"> </span><span class="c1">// Free memory</span>
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<a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w"> </span><span class="k">delete</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
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<a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w"> </span><span class="c1">// Number of child nodes = 2</span>
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<a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w"> </span><span class="c1">// Get the next node in in-order traversal of cur</span>
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<a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">right</span><span class="p">;</span>
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<a id="__codelineno-29-44" name="__codelineno-29-44" href="#__codelineno-29-44"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-></span><span class="n">left</span><span class="p">;</span>
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<a id="__codelineno-29-46" name="__codelineno-29-46" href="#__codelineno-29-46"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">tmpVal</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-></span><span class="n">val</span><span class="p">;</span>
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<a id="__codelineno-29-48" name="__codelineno-29-48" href="#__codelineno-29-48"></a><span class="w"> </span><span class="c1">// Recursively remove node tmp</span>
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<a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="o">-></span><span class="n">val</span><span class="p">);</span>
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<a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w"> </span><span class="c1">// Replace cur with tmp</span>
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<a id="__codelineno-29-51" name="__codelineno-29-51" href="#__codelineno-29-51"></a><span class="w"> </span><span class="n">cur</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmpVal</span><span class="p">;</span>
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<a id="__codelineno-29-52" name="__codelineno-29-52" href="#__codelineno-29-52"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-29-53" name="__codelineno-29-53" href="#__codelineno-29-53"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Remove node */</span>
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<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w"> </span><span class="c1">// If tree is empty, return</span>
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<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
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<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
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<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
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<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w"> </span><span class="c1">// Loop find, break after passing leaf nodes</span>
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<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w"> </span><span class="c1">// Found node to be removed, break loop</span>
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<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
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<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w"> </span><span class="k">break</span><span class="p">;</span>
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<a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
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<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w"> </span><span class="c1">// Node to be removed is in cur's right subtree</span>
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|
|
<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
|
|
|
<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w"> </span><span class="c1">// Node to be removed is in cur's left subtree</span>
|
|
|
<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w"> </span><span class="k">else</span>
|
|
|
<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w"> </span><span class="c1">// If no node to be removed, return</span>
|
|
|
<a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
|
|
|
<a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w"> </span><span class="k">return</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w"> </span><span class="c1">// Number of child nodes = 0 or 1</span>
|
|
|
<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="w"> </span><span class="c1">// When the number of child nodes = 0/1, child = null/that child node</span>
|
|
|
<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w"> </span><span class="c1">// Remove node cur</span>
|
|
|
<a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
|
|
|
<a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w"> </span><span class="k">else</span>
|
|
|
<a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w"> </span><span class="c1">// If the removed node is the root, reassign the root</span>
|
|
|
<a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="w"> </span><span class="c1">// Number of child nodes = 2</span>
|
|
|
<a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w"> </span><span class="c1">// Get the next node in in-order traversal of cur</span>
|
|
|
<a id="__codelineno-30-41" name="__codelineno-30-41" href="#__codelineno-30-41"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-42" name="__codelineno-30-42" href="#__codelineno-30-42"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
|
|
<a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a><span class="w"> </span><span class="c1">// Recursively remove node tmp</span>
|
|
|
<a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
|
|
|
<a id="__codelineno-30-47" name="__codelineno-30-47" href="#__codelineno-30-47"></a><span class="w"> </span><span class="c1">// Replace cur with tmp</span>
|
|
|
<a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
|
|
|
<a id="__codelineno-30-49" name="__codelineno-30-49" href="#__codelineno-30-49"></a><span class="w"> </span><span class="p">}</span>
|
|
|
<a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
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<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">Remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{</span><span class="nx">binarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
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|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
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|
|
</div>
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|
<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
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|
|
</div>
|
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|
<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{</span><span class="nx">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
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|
<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
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|
<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">removeItem</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
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|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="o">[</span><span class="n">class</span><span class="o">]</span><span class="p">{</span><span class="no">BinarySearchTree</span><span class="p">}</span><span class="o">-[</span><span class="n">func</span><span class="o">]</span><span class="p">{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_search_tree.zig</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{</span><span class="n">BinarySearchTree</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">remove</span><span class="p">}</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
</div>
|
|
|
</div>
|
|
|
<h3 id="4-in-order-traversal-is-ordered">4. In-order traversal is ordered<a class="headerlink" href="#4-in-order-traversal-is-ordered" title="Permanent link">¶</a></h3>
|
|
|
<p>As shown in Figure 7-22, the in-order traversal of a binary tree follows the traversal order of "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right," and a binary search tree satisfies the size relationship of "left child node <span class="arithmatex">\(<\)</span> root node <span class="arithmatex">\(<\)</span> right child node."</p>
|
|
|
<p>This means that when performing in-order traversal in a binary search tree, the next smallest node will always be traversed first, thus leading to an important property: <strong>The sequence of in-order traversal in a binary search tree is ascending</strong>.</p>
|
|
|
<p>Using the ascending property of in-order traversal, obtaining ordered data in a binary search tree requires only <span class="arithmatex">\(O(n)\)</span> time, without the need for additional sorting operations, which is very efficient.</p>
|
|
|
<p><a class="glightbox" href="../binary_search_tree.assets/bst_inorder_traversal.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="In-order traversal sequence of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></a></p>
|
|
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<p align="center"> Figure 7-22 In-order traversal sequence of a binary search tree </p>
|
|
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|
|
|
<h2 id="742-efficiency-of-binary-search-trees">7.4.2 Efficiency of binary search trees<a class="headerlink" href="#742-efficiency-of-binary-search-trees" title="Permanent link">¶</a></h2>
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|
|
<p>Given a set of data, we consider using an array or a binary search tree for storage. Observing Table 7-2, the operations on a binary search tree all have logarithmic time complexity, which is stable and efficient. Arrays are more efficient than binary search trees only in scenarios involving frequent additions and infrequent searches or removals.</p>
|
|
|
<p align="center"> Table 7-2 Efficiency comparison between arrays and search trees </p>
|
|
|
|
|
|
<div class="center-table">
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th></th>
|
|
|
<th>Unsorted array</th>
|
|
|
<th>Binary search tree</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Search element</td>
|
|
|
<td><span class="arithmatex">\(O(n)\)</span></td>
|
|
|
<td><span class="arithmatex">\(O(\log n)\)</span></td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>Insert element</td>
|
|
|
<td><span class="arithmatex">\(O(1)\)</span></td>
|
|
|
<td><span class="arithmatex">\(O(\log n)\)</span></td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>Remove element</td>
|
|
|
<td><span class="arithmatex">\(O(n)\)</span></td>
|
|
|
<td><span class="arithmatex">\(O(\log n)\)</span></td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</div>
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|
|
<p>Ideally, the binary search tree is "balanced," allowing any node can be found within <span class="arithmatex">\(\log n\)</span> loops.</p>
|
|
|
<p>However, if we continuously insert and remove nodes in a binary search tree, it may degenerate into a linked list as shown in Figure 7-23, where the time complexity of various operations also degrades to <span class="arithmatex">\(O(n)\)</span>.</p>
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<p><a class="glightbox" href="../binary_search_tree.assets/bst_degradation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Degradation of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_degradation.png" /></a></p>
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<p align="center"> Figure 7-23 Degradation of a binary search tree </p>
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<h2 id="743-common-applications-of-binary-search-trees">7.4.3 Common applications of binary search trees<a class="headerlink" href="#743-common-applications-of-binary-search-trees" title="Permanent link">¶</a></h2>
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<ul>
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<li>Used as multi-level indexes in systems to implement efficient search, insertion, and removal operations.</li>
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<li>Serves as the underlying data structure for certain search algorithms.</li>
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<li>Used to store data streams to maintain their ordered state.</li>
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