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krahets/hello-algo
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</div>
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</a>
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</div>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
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0. 写在前面
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_1">
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<span class="md-nav__icon md-icon"></span>
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0. 写在前面
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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_preface/about_the_book/" class="md-nav__link">
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0.1. 关于本书
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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_preface/suggestions/" class="md-nav__link">
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0.2. 如何使用本书
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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_preface/summary/" class="md-nav__link">
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0.3. 小结
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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 md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
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1. 引言
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_2">
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<span class="md-nav__icon md-icon"></span>
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1. 引言
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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_introduction/algorithms_are_everywhere/" class="md-nav__link">
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1.1. 算法无处不在
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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_introduction/what_is_dsa/" class="md-nav__link">
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1.2. 算法是什么
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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_introduction/summary/" class="md-nav__link">
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1.3. 小结
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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 md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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2. 复杂度分析
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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2. 复杂度分析
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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_computational_complexity/performance_evaluation/" class="md-nav__link">
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2.1. 算法效率评估
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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_computational_complexity/time_complexity/" class="md-nav__link">
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2.2. 时间复杂度
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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_computational_complexity/space_complexity/" class="md-nav__link">
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2.3. 空间复杂度
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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_computational_complexity/space_time_tradeoff/" class="md-nav__link">
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2.4. 权衡时间与空间
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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_computational_complexity/summary/" class="md-nav__link">
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2.5. 小结
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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 md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
|
|
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3. 数据结构简介
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|
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<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
|
|
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
|
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3. 数据结构简介
|
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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_data_structure/data_and_memory/" class="md-nav__link">
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3.1. 数据与内存
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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_data_structure/classification_of_data_structure/" class="md-nav__link">
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3.2. 数据结构分类
|
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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_data_structure/summary/" class="md-nav__link">
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3.3. 小结
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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 md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
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4. 数组与链表
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
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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_array_and_linkedlist/array/" class="md-nav__link">
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4.1. 数组
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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_array_and_linkedlist/linked_list/" class="md-nav__link">
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4.2. 链表
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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_array_and_linkedlist/list/" class="md-nav__link">
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4.3. 列表
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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_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 小结
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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 md-nav__item--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
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5. 栈与队列
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
|
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5. 栈与队列
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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_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈
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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_stack_and_queue/queue/" class="md-nav__link">
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5.2. 队列
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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_stack_and_queue/deque/" class="md-nav__link">
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5.3. 双向队列
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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_stack_and_queue/summary/" class="md-nav__link">
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5.4. 小结
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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 md-nav__item--section md-nav__item--nested">
|
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|
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
|
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
|
|
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6. 散列表
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_7">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
6. 散列表
|
|
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</label>
|
|
|
<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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6.1. 哈希表
|
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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_hashing/hash_collision/" class="md-nav__link">
|
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6.2. 哈希冲突处理
|
|
|
</a>
|
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</li>
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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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6.3. 小结
|
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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 md-nav__item--active md-nav__item--section md-nav__item--nested">
|
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
|
|
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7. 树
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
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|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
|
|
|
<label class="md-nav__title" for="__nav_8">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
7. 树
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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 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">
|
|
|
7.1. 二叉树
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
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|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|
|
7.1. 二叉树
|
|
|
</a>
|
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|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
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|
<label class="md-nav__title" for="__toc">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
目录
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#711" class="md-nav__link">
|
|
|
7.1.1. 二叉树常见术语
|
|
|
</a>
|
|
|
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#712" class="md-nav__link">
|
|
|
7.1.2. 二叉树基本操作
|
|
|
</a>
|
|
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|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="#713" class="md-nav__link">
|
|
|
7.1.3. 常见二叉树类型
|
|
|
</a>
|
|
|
|
|
|
<nav class="md-nav" aria-label="7.1.3. 常见二叉树类型">
|
|
|
<ul class="md-nav__list">
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#_1" class="md-nav__link">
|
|
|
完美二叉树
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#_2" class="md-nav__link">
|
|
|
完全二叉树
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#_3" class="md-nav__link">
|
|
|
完满二叉树
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#_4" class="md-nav__link">
|
|
|
平衡二叉树
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
</ul>
|
|
|
</nav>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#714" class="md-nav__link">
|
|
|
7.1.4. 二叉树的退化
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="#715" class="md-nav__link">
|
|
|
7.1.5. 二叉树表示方式 *
|
|
|
</a>
|
|
|
|
|
|
</li>
|
|
|
|
|
|
</ul>
|
|
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|
|
|
</nav>
|
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|
|
</li>
|
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|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../binary_tree_traversal/" class="md-nav__link">
|
|
|
7.2. 二叉树遍历
|
|
|
</a>
|
|
|
</li>
|
|
|
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|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../binary_search_tree/" class="md-nav__link">
|
|
|
7.3. 二叉搜索树
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../avl_tree/" class="md-nav__link">
|
|
|
7.4. AVL 树 *
|
|
|
</a>
|
|
|
</li>
|
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<li class="md-nav__item">
|
|
|
<a href="../summary/" class="md-nav__link">
|
|
|
7.5. 小结
|
|
|
</a>
|
|
|
</li>
|
|
|
|
|
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|
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|
|
|
|
</ul>
|
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|
</nav>
|
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</li>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
|
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|
|
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
|
|
|
8. 堆
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
8. 堆
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
|
8.1. 堆
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
|
|
8.2. 建堆操作 *
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_heap/summary/" class="md-nav__link">
|
|
|
8.3. 小结
|
|
|
</a>
|
|
|
</li>
|
|
|
|
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|
|
</ul>
|
|
|
</nav>
|
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</li>
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|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
|
|
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|
|
|
|
|
|
|
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|
<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
|
|
|
9. 图
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_10">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
9. 图
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
|
|
9.1. 图
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
|
|
9.2. 图基础操作
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
|
|
9.3. 图的遍历
|
|
|
</a>
|
|
|
</li>
|
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|
|
|
|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
|
|
9.4. 小结
|
|
|
</a>
|
|
|
</li>
|
|
|
|
|
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|
|
|
|
</ul>
|
|
|
</nav>
|
|
|
</li>
|
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|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
|
|
|
10. 查找算法
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
</label>
|
|
|
|
|
|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_11">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
10. 查找算法
|
|
|
</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|
|
10.1. 线性查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|
|
10.2. 二分查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
|
|
10.3. 哈希查找
|
|
|
</a>
|
|
|
</li>
|
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|
|
<li class="md-nav__item">
|
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
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10.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
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11. 排序算法
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<span class="md-nav__icon md-icon"></span>
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</label>
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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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11. 排序算法
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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_sorting/sorting_algorithm/" class="md-nav__link">
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11.1. 排序算法
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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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11.2. 冒泡排序
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</a>
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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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11.3. 插入排序
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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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11.4. 快速排序
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</a>
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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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11.5. 归并排序
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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/bucket_sort/" class="md-nav__link">
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11.6. 桶排序（New）
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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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11.7. 计数排序（New）
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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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11.8. 基数排序（New）
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</a>
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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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11.9. 小结
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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--section md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
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12. 回溯算法
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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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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12. 回溯算法
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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12.1. 回溯算法（New）
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
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13. 附录
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<span class="md-nav__icon md-icon"></span>
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</label>
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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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13. 附录
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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_appendix/installation/" class="md-nav__link">
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13.1. 编程环境安装
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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_appendix/contribution/" class="md-nav__link">
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13.2. 一起参与创作
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</a>
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</li>
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</ul>
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</nav>
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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__link--index ">
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<a href="../../chapter_reference/">参考文献</a>
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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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参考文献
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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</nav>
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</ul>
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</nav>
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</div>
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</div>
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</div>
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<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
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<div class="md-sidebar__scrollwrap">
|
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<div class="md-sidebar__inner">
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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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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目录
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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="#711" class="md-nav__link">
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7.1.1. 二叉树常见术语
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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="#712" class="md-nav__link">
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7.1.2. 二叉树基本操作
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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="#713" class="md-nav__link">
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7.1.3. 常见二叉树类型
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</a>
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<nav class="md-nav" aria-label="7.1.3. 常见二叉树类型">
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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" class="md-nav__link">
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完美二叉树
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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" class="md-nav__link">
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完全二叉树
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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" class="md-nav__link">
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完满二叉树
|
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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" class="md-nav__link">
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平衡二叉树
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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="#714" class="md-nav__link">
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7.1.4. 二叉树的退化
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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="#715" class="md-nav__link">
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7.1.5. 二叉树表示方式 *
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</a>
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</li>
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</ul>
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</nav>
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</div>
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</div>
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</div>
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<div class="md-content" data-md-component="content">
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<article class="md-content__inner md-typeset">
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<a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_tree/binary_tree.md" title="编辑此页" class="md-content__button md-icon">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
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</a>
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<h1 id="71">7.1. 二叉树<a class="headerlink" href="#71" title="Permanent link">¶</a></h1>
|
|
|
<p>「二叉树 Binary Tree」是一种非线性数据结构，代表着祖先与后代之间的派生关系，体现着“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含一个「值」和两个「指针」。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><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" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">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"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 二叉树节点类 */</span>
|
|
|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
|
|
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
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<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
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<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></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"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 二叉树节点结构体 */</span>
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<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</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="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
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<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="k">nullptr</span><span class="p">),</span><span class="w"> </span><span class="n">right</span><span class="p">(</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-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></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"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">class</span> <span class="nc">TreeNode</span><span class="p">:</span>
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<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">"""二叉树节点类"""</span>
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<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</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="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span> <span class="c1"># 节点值</span>
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<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 左子节点指针</span>
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<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 右子节点指针</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"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 二叉树节点结构体 */</span>
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<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="nx">Val</span><span class="w"> </span><span class="kt">int</span>
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<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">Left</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span>
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<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w"> </span><span class="nx">Right</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span>
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<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="p">}</span>
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<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* 节点初始化方法 */</span>
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<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">&</span><span class="nx">TreeNode</span><span class="p">{</span>
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<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="nx">Left</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span>
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<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="nx">Right</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span>
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<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="nx">Val</span><span class="p">:</span><span class="w"> </span><span class="nx">v</span><span class="p">,</span>
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<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></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"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 二叉树节点类 */</span>
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<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// 节点值</span>
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<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</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="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
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<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</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="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
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<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></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"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 二叉树节点类 */</span>
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<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
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<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</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-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</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-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>
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<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">left?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</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="nx">right?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</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-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
|
|
|
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
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<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
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<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></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"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
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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"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 二叉树节点类 */</span>
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<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点值</span>
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<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子节点指针</span>
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<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子节点指针</span>
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<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></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"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 二叉树节点类 */</span>
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<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span> <span class="p">{</span>
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<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="kd">var</span> <span class="nv">val</span><span class="p">:</span> <span class="nb">Int</span> <span class="c1">// 节点值</span>
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<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a> <span class="kd">var</span> <span class="nv">left</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 左子节点指针</span>
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<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">right</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">?</span> <span class="c1">// 右子节点指针</span>
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<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
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<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span>
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<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="n">val</span> <span class="p">=</span> <span class="n">x</span>
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<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a> <span class="p">}</span>
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<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></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"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a>
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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>节点的两个指针分别指向「左子节点」和「右子节点」，同时该节点被称为这两个子节点的「父节点」。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的「左子树」，同理可得「右子树」。</p>
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<p><strong>在二叉树中，除叶节点外，其他所有节点都包含子节点和非空子树</strong>。例如，在以下示例中，若将“节点 2”视为父节点，则其左子节点和右子节点分别是“节点 4”和“节点 5”，左子树是“节点 4 及其以下节点形成的树”，右子树是“节点 5 及其以下节点形成的树”。</p>
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<p><img alt="父节点、子节点、子树" src="../binary_tree.assets/binary_tree_definition.png" /></p>
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<p align="center"> Fig. 父节点、子节点、子树 </p>
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<h2 id="711">7.1.1. 二叉树常见术语<a class="headerlink" href="#711" title="Permanent link">¶</a></h2>
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<p>二叉树涉及的术语较多，建议尽量理解并记住。</p>
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<ul>
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<li>「根节点 Root Node」：位于二叉树顶层的节点，没有父节点；</li>
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<li>「叶节点 Leaf Node」：没有子节点的节点，其两个指针均指向 <span class="arithmatex">\(\text{null}\)</span> ；</li>
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<li>节点的「层 Level」：从顶至底递增，根节点所在层为 1 ；</li>
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<li>节点的「度 Degree」：节点的子节点的数量。在二叉树中，度的范围是 0, 1, 2 ；</li>
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<li>「边 Edge」：连接两个节点的线段，即节点指针；</li>
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<li>二叉树的「高度」：从根节点到最远叶节点所经过的边的数量；</li>
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<li>节点的「深度 Depth」 ：从根节点到该节点所经过的边的数量；</li>
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<li>节点的「高度 Height」：从最远叶节点到该节点所经过的边的数量；</li>
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</ul>
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<p><img alt="二叉树的常用术语" src="../binary_tree.assets/binary_tree_terminology.png" /></p>
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<p align="center"> Fig. 二叉树的常用术语 </p>
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<div class="admonition tip">
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<p class="admonition-title">高度与深度的定义</p>
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<p>请注意，我们通常将「高度」和「深度」定义为“走过边的数量”，但有些题目或教材可能会将其定义为“走过节点的数量”。在这种情况下，高度和深度都需要加 1 。</p>
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</div>
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<h2 id="712">7.1.2. 二叉树基本操作<a class="headerlink" href="#712" title="Permanent link">¶</a></h2>
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<p><strong>初始化二叉树</strong>。与链表类似，首先初始化节点，然后构建引用指向（即指针）。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">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_tree.java</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n1</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="mi">1</span><span class="p">);</span>
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n2</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="mi">2</span><span class="p">);</span>
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<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n3</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="mi">3</span><span class="p">);</span>
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<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n4</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="mi">4</span><span class="p">);</span>
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<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n5</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="mi">5</span><span class="p">);</span>
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<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="n">n1</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">n2</span><span class="p">;</span>
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<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="n">n1</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">n3</span><span class="p">;</span>
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<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="n">n2</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">n4</span><span class="p">;</span>
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<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="n">n2</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">n5</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_tree.cpp</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</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="mi">1</span><span class="p">);</span>
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<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</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="mi">2</span><span class="p">);</span>
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<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n3</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="mi">3</span><span class="p">);</span>
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<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n4</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="mi">4</span><span class="p">);</span>
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<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n5</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="mi">5</span><span class="p">);</span>
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<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">n1</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">n2</span><span class="p">;</span>
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<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="n">n1</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">n3</span><span class="p">;</span>
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<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="n">n2</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">n4</span><span class="p">;</span>
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<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="n">n2</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">n5</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_tree.py</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># 初始化二叉树</span>
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<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># 初始化节点</span>
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<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">n1</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="n">n2</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="n">n3</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
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<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="n">n4</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
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<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="n">n5</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
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<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="c1"># 构建引用指向（即指针）</span>
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<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</span>
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<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">n1</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">n3</span>
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<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="n">n2</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n4</span>
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<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="n">n2</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">n5</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_tree.go</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="nx">n1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="nx">n2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
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<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="nx">n3</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
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<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="nx">n4</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
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<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="nx">n5</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
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<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n2</span>
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<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n3</span>
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<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n4</span>
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<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n5</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_tree.js</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">1</span><span class="p">),</span>
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<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w"> </span><span class="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">2</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="w"> </span><span class="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">3</span><span class="p">),</span>
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<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w"> </span><span class="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">4</span><span class="p">),</span>
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<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w"> </span><span class="nx">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
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<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</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="nx">n1</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span><span class="p">;</span>
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<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n4</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="nx">n2</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n5</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_tree.ts</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">1</span><span class="p">),</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="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">2</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="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">3</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="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">4</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="nx">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">5</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="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
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<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</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="nx">n2</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n4</span><span class="p">;</span>
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<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n5</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_tree.c</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a>
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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_tree.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n1</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="m">1</span><span class="p">);</span>
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<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n2</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="m">2</span><span class="p">);</span>
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<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n3</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="m">3</span><span class="p">);</span>
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<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n4</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="m">4</span><span class="p">);</span>
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<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n5</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="m">5</span><span class="p">);</span>
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<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
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<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
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<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
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<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="n">n2</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n5</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_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">let</span> <span class="nv">n1</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span>
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<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="kd">let</span> <span class="nv">n2</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="p">)</span>
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<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="kd">let</span> <span class="nv">n3</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">3</span><span class="p">)</span>
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<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="kd">let</span> <span class="nv">n4</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">4</span><span class="p">)</span>
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<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="kd">let</span> <span class="nv">n5</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">5</span><span class="p">)</span>
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<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n2</span>
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<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="n">n1</span><span class="p">.</span><span class="kr">right</span> <span class="p">=</span> <span class="n">n3</span>
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<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="n">n2</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n4</span>
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<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="n">n2</span><span class="p">.</span><span class="kr">right</span> <span class="p">=</span> <span class="n">n5</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_tree.zig</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a>
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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><strong>插入与删除节点</strong>。与链表类似，通过修改指针来实现插入与删除节点。</p>
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<p><img alt="在二叉树中插入与删除节点" src="../binary_tree.assets/binary_tree_add_remove.png" /></p>
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<p align="center"> Fig. 在二叉树中插入与删除节点 </p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:10"><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" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">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_tree.java</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">P</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="mi">0</span><span class="p">);</span>
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<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="n">n1</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">P</span><span class="p">;</span>
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<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="n">P</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">n2</span><span class="p">;</span>
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<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="n">n1</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">n2</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_tree.cpp</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">P</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="mi">0</span><span class="p">);</span>
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<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="n">n1</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">P</span><span class="p">;</span>
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<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="n">P</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">n2</span><span class="p">;</span>
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<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="c1">// 删除节点 P</span>
|
|
|
<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="n">n1</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">n2</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_tree.py</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="c1"># 插入与删除节点</span>
|
|
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<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="n">p</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|
|
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="c1"># 在 n1 -> n2 中间插入节点 P</span>
|
|
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<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">p</span>
|
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|
<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="n">p</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</span>
|
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<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="c1"># 删除节点 P</span>
|
|
|
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</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_tree.go</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 插入与删除节点 */</span>
|
|
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<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
|
|
|
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
|
|
<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">p</span>
|
|
|
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="nx">p</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n2</span>
|
|
|
<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="c1">// 删除节点 P</span>
|
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<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">n2</span>
|
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</code></pre></div>
|
|
|
</div>
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|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.js</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 插入与删除节点 */</span>
|
|
|
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
|
|
|
<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
|
|
|
<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">P</span><span class="p">;</span>
|
|
|
<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="nx">P</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
|
|
|
<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="c1">// 删除节点 P</span>
|
|
|
<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
|
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|
</code></pre></div>
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|
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</div>
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<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.ts</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="cm">/* 插入与删除节点 */</span>
|
|
|
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
|
|
|
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
|
|
|
<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">P</span><span class="p">;</span>
|
|
|
<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="nx">P</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
|
|
|
<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1">// 删除节点 P</span>
|
|
|
<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.c</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a>
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</code></pre></div>
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</div>
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<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.cs</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* 插入与删除节点 */</span>
|
|
|
<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">P</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="m">0</span><span class="p">);</span>
|
|
|
<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
|
|
|
<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
|
|
|
<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="n">P</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
|
|
|
<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="c1">// 删除节点 P</span>
|
|
|
<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
|
|
|
</code></pre></div>
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|
</div>
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|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.swift</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="kd">let</span> <span class="nv">P</span> <span class="p">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
|
|
|
<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
|
|
|
<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">P</span>
|
|
|
<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="n">P</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n2</span>
|
|
|
<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="c1">// 删除节点 P</span>
|
|
|
<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n2</span>
|
|
|
</code></pre></div>
|
|
|
</div>
|
|
|
<div class="tabbed-block">
|
|
|
<div class="highlight"><span class="filename">binary_tree.zig</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a>
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|
|
</code></pre></div>
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|
|
</div>
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|
</div>
|
|
|
</div>
|
|
|
<div class="admonition note">
|
|
|
<p class="admonition-title">Note</p>
|
|
|
<p>需要注意的是，插入节点可能会改变二叉树的原有逻辑结构，而删除节点通常意味着删除该节点及其所有子树。因此，在二叉树中，插入与删除操作通常是由一套操作配合完成的，以实现有实际意义的操作。</p>
|
|
|
</div>
|
|
|
<h2 id="713">7.1.3. 常见二叉树类型<a class="headerlink" href="#713" title="Permanent link">¶</a></h2>
|
|
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<h3 id="_1">完美二叉树<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
|
|
<p>「完美二叉树 Perfect Binary Tree」除了最底层外，其余所有层的节点都被完全填满。在完美二叉树中，叶节点的度为 <span class="arithmatex">\(0\)</span> ，其余所有节点的度都为 <span class="arithmatex">\(2\)</span> ；若树高度为 <span class="arithmatex">\(h\)</span> ，则节点总数为 <span class="arithmatex">\(2^{h+1} - 1\)</span> ，呈现标准的指数级关系，反映了自然界中常见的细胞分裂现象。</p>
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|
|
<div class="admonition tip">
|
|
|
<p class="admonition-title">Tip</p>
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|
|
<p>在中文社区中，完美二叉树常被称为「满二叉树」，请注意区分。</p>
|
|
|
</div>
|
|
|
<p><img alt="完美二叉树" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
|
|
|
<p align="center"> Fig. 完美二叉树 </p>
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|
<h3 id="_2">完全二叉树<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
|
|
<p>「完全二叉树 Complete Binary Tree」只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
|
|
|
<p><img alt="完全二叉树" src="../binary_tree.assets/complete_binary_tree.png" /></p>
|
|
|
<p align="center"> Fig. 完全二叉树 </p>
|
|
|
|
|
|
<h3 id="_3">完满二叉树<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
|
|
<p>「完满二叉树 Full Binary Tree」除了叶节点之外，其余所有节点都有两个子节点。</p>
|
|
|
<p><img alt="完满二叉树" src="../binary_tree.assets/full_binary_tree.png" /></p>
|
|
|
<p align="center"> Fig. 完满二叉树 </p>
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|
<h3 id="_4">平衡二叉树<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
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<p>「平衡二叉树 Balanced Binary Tree」中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
|
|
|
<p><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
|
|
|
<p align="center"> Fig. 平衡二叉树 </p>
|
|
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|
|
<h2 id="714">7.1.4. 二叉树的退化<a class="headerlink" href="#714" title="Permanent link">¶</a></h2>
|
|
|
<p>当二叉树的每层节点都被填满时，达到「完美二叉树」；而当所有节点都偏向一侧时，二叉树退化为「链表」。</p>
|
|
|
<ul>
|
|
|
<li>完美二叉树是理想情况，可以充分发挥二叉树“分治”的优势；</li>
|
|
|
<li>链表则是另一个极端，各项操作都变为线性操作，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> ；</li>
|
|
|
</ul>
|
|
|
<p><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_corner_cases.png" /></p>
|
|
|
<p align="center"> Fig. 二叉树的最佳与最差结构 </p>
|
|
|
|
|
|
<p>如下表所示，在最佳和最差结构下，二叉树的叶节点数量、节点总数、高度等达到极大或极小值。</p>
|
|
|
<div class="center-table">
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th></th>
|
|
|
<th>完美二叉树</th>
|
|
|
<th>链表</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>第 <span class="arithmatex">\(i\)</span> 层的节点数量</td>
|
|
|
<td><span class="arithmatex">\(2^{i-1}\)</span></td>
|
|
|
<td><span class="arithmatex">\(1\)</span></td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>树的高度为 <span class="arithmatex">\(h\)</span> 时的叶节点数量</td>
|
|
|
<td><span class="arithmatex">\(2^h\)</span></td>
|
|
|
<td><span class="arithmatex">\(1\)</span></td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>树的高度为 <span class="arithmatex">\(h\)</span> 时的节点总数</td>
|
|
|
<td><span class="arithmatex">\(2^{h+1} - 1\)</span></td>
|
|
|
<td><span class="arithmatex">\(h + 1\)</span></td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>树的节点总数为 <span class="arithmatex">\(n\)</span> 时的高度</td>
|
|
|
<td><span class="arithmatex">\(\log_2 (n+1) - 1\)</span></td>
|
|
|
<td><span class="arithmatex">\(n - 1\)</span></td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</div>
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|
<h2 id="715">7.1.5. 二叉树表示方式 *<a class="headerlink" href="#715" title="Permanent link">¶</a></h2>
|
|
|
<p>我们通常使用二叉树的「链表表示」，即存储单位为节点 <code>TreeNode</code> ，节点之间通过指针相连接。本文前述示例代码展示了二叉树在链表表示下的各项基本操作。</p>
|
|
|
<p>那么，能否用「数组」来表示二叉树呢？答案是肯定的。先来分析一个简单案例，给定一个「完美二叉树」，将节点按照层序遍历的顺序编号（从 0 开始），那么可以推导得出父节点索引与子节点索引之间的“映射公式”：<strong>若节点的索引为 <span class="arithmatex">\(i\)</span> ，则该节点的左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span></strong> 。</p>
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<p><strong>本质上，映射公式的作用相当于链表中的指针</strong>。对于层序遍历序列中的任意节点，我们都可以使用映射公式来访问其子节点。因此，我们可以将二叉树的层序遍历序列存储到数组中，利用以上映射公式来表示二叉树。</p>
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<p><img alt="完美二叉树的数组表示" src="../binary_tree.assets/array_representation_mapping.png" /></p>
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<p align="center"> Fig. 完美二叉树的数组表示 </p>
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<p>然而，完美二叉树只是一个特例。在二叉树的中间层，通常存在许多 <span class="arithmatex">\(\text{null}\)</span> ，而层序遍历序列并不包含这些 <span class="arithmatex">\(\text{null}\)</span> 。我们无法仅凭序列来推测空节点的数量和分布位置，<strong>这意味着理论上存在许多种二叉树都符合该层序遍历序列</strong>。显然，在这种情况下，我们无法使用数组来存储二叉树。</p>
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<p><img alt="给定数组对应多种二叉树可能性" src="../binary_tree.assets/array_representation_without_empty.png" /></p>
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<p align="center"> Fig. 给定数组对应多种二叉树可能性 </p>
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<p>为了解决这个问题，我们可以考虑按照完美二叉树的形式来表示所有二叉树，<strong>并在序列中使用特殊符号来显式地表示 <span class="arithmatex">\(\text{null}\)</span></strong>。如下图所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:10"><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" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">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"><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="c1">// 使用 int 的包装类 Integer ，就可以使用 null 来标记空位</span>
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<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="w"> </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"><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="c1">// 为了符合数据类型为 int ，使用 int 最大值标记空位</span>
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<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="c1">// 该方法的使用前提是没有节点的值 = INT_MAX</span>
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<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">></span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="w"> </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"><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="c1"># 二叉树的数组表示</span>
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<a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="c1"># 直接使用 None 来表示空位</span>
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<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="n">tree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">15</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"><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="c1">// 使用 any 类型的切片, 就可以使用 nil 来标记空位</span>
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<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="nx">tree</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">15</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"><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="c1">// 直接使用 null 来表示空位</span>
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<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</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"><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="c1">// 直接使用 null 来表示空位</span>
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<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="kt">number</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="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</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"><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a>
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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"><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="c1">// 使用 int? 可空类型 ，就可以使用 null 来标记空位</span>
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<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="kt">int?</span><span class="p">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="w"> </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"><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="cm">/* 二叉树的数组表示 */</span>
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<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="c1">// 使用 Int? 可空类型 ，就可以使用 nil 来标记空位</span>
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<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="kd">let</span> <span class="nv">tree</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">?]</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="kc">nil</span><span class="p">,</span> <span class="mi">15</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"><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a>
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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><img alt="任意类型二叉树的数组表示" src="../binary_tree.assets/array_representation_with_empty.png" /></p>
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<p align="center"> Fig. 任意类型二叉树的数组表示 </p>
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<p><strong>完全二叉树非常适合使用数组来表示</strong>。回顾「完全二叉树」的定义，<span class="arithmatex">\(\text{null}\)</span> 只出现在最底层，并且最底层的节点尽量靠左。这意味着，<strong>所有空节点一定出现在层序遍历序列的末尾</strong>。由于我们事先知道了所有 <span class="arithmatex">\(\text{null}\)</span> 的位置，因此在使用数组表示完全二叉树时，可以省略存储它们。</p>
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<p><img alt="完全二叉树的数组表示" src="../binary_tree.assets/array_representation_complete_binary_tree.png" /></p>
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<p align="center"> Fig. 完全二叉树的数组表示 </p>
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<p>数组表示具有两个显著优点：首先，它不需要存储指针，从而节省了空间；其次，它允许随机访问节点。然而，当二叉树中存在大量 <span class="arithmatex">\(\text{null}\)</span> 时，数组中包含的节点数据比重较低，导致有效空间利用率降低。</p>
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<h2 id="__comments">评论</h2>
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