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krahets/hello-algo
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第 0 章 前言
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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第 1 章 初识算法
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1.1 算法无处不在
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1.2 算法是什么
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1.3 小结
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第 2 章 复杂度分析
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第 2 章 复杂度分析
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2.1 算法效率评估
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2.2 迭代与递归
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2.3 时间复杂度
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2.4 空间复杂度
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2.5 小结
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第 3 章 数据结构
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<span class="md-nav__icon md-icon"></span>
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第 3 章 数据结构
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1 数据结构分类
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<span class="md-ellipsis">
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3.2 基本数据类型
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<span class="md-ellipsis">
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3.3 数字编码 *
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<span class="md-ellipsis">
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3.4 字符编码 *
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3.5 小结
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
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<span class="md-ellipsis">
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第 4 章 数组与链表
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</a>
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<label class="md-nav__link " for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 4 章 数组与链表
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1 数组
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</a>
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 链表
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 列表
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 小结
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
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<span class="md-ellipsis">
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第 5 章 栈与队列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 5 章 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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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 栈
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 队列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 双向队列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 小结
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</span>
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</a>
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</li>
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</nav>
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<span class="md-ellipsis">
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第 6 章 哈希表
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</a>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 6 章 哈希表
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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_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1 哈希表
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 哈希冲突
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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6.3 哈希算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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6.4 小结
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</span>
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</a>
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</li>
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</nav>
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<span class="md-ellipsis">
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第 7 章 树
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第 7 章 树
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7.1 二叉树
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7.1 二叉树
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7.1.1 二叉树常见术语
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1. 初始化二叉树
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2. 插入与删除节点
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7.1.3 常见二叉树类型
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1. 完美二叉树
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2. 完全二叉树
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3. 完满二叉树
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4. 平衡二叉树
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7.1.4 二叉树的退化
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7.2 二叉树遍历
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7.3 二叉树数组表示
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7.4 二叉搜索树
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7.5 AVL 树 *
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7.6 小结
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第 8 章 堆
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第 8 章 堆
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8.1 堆
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8.2 建堆操作
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8.3 Top-K 问题
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8.4 小结
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</a>
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<a href="../../chapter_graph/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
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<span class="md-ellipsis">
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第 9 章 图
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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第 9 章 图
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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_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1 图
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2 图基础操作
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 图的遍历
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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9.4 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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第 10 章 搜索
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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第 10 章 搜索
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
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10.1 二分查找
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 二分查找插入点
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3 二分查找边界
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 哈希优化策略
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 重识搜索算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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|
10.6 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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|
<div class="md-nav__link md-nav__container">
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<a href="../../chapter_sorting/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
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<span class="md-ellipsis">
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第 11 章 排序
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|
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</span>
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</a>
|
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|
<label class="md-nav__link " for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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|
|
<label class="md-nav__title" for="__nav_12">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
第 11 章 排序
|
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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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<span class="md-ellipsis">
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11.1 排序算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.2 选择排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.3 冒泡排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.4 插入排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.5 快速排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.6 归并排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
11.7 堆排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.8 桶排序
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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|
11.9 计数排序
|
|
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</span>
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<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.10 基数排序
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<li class="md-nav__item">
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<a href="../../chapter_sorting/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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11.11 小结
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</a>
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第 12 章 分治
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<label class="md-nav__link " for="__nav_13">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_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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第 12 章 分治
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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12.1 分治算法
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</span>
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
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12.2 分治搜索策略
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</span>
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<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.3 构建树问题
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</span>
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<span class="md-status md-status--new" title="最近添加">
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 汉诺塔问题
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</span>
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<span class="md-status md-status--new" title="最近添加">
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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12.5 小结
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</span>
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<span class="md-status md-status--new" title="最近添加">
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<a href="../../chapter_backtracking/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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<span class="md-ellipsis">
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第 13 章 回溯
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</span>
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</a>
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<label class="md-nav__link " for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
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第 13 章 回溯
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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13.1 回溯算法
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</span>
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</a>
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.2 全排列问题
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3 子集和问题
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.4 N 皇后问题
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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13.5 小结
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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<a href="../../chapter_dynamic_programming/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 14 章 动态规划
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</span>
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<span class="md-status md-status--new" title="最近添加">
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</a>
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_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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第 14 章 动态规划
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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_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
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<span class="md-ellipsis">
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14.1 初探动态规划
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</span>
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<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
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<span class="md-ellipsis">
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14.2 DP 问题特性
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</span>
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14.3 DP 解题思路
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14.4 0-1 背包问题
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14.5 完全背包问题
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14.6 编辑距离问题
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14.7 小结
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第 15 章 贪心
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第 15 章 贪心
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15.1 贪心算法
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15.2 分数背包问题
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15.3 最大容量问题
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15.4 最大切分乘积问题
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15.5 小结
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第 16 章 附录
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16.1 编程环境安装
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16.2 一起参与创作
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参考文献
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<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
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7.1.1 二叉树常见术语
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1. 初始化二叉树
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<li class="md-nav__item">
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2. 插入与删除节点
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7.1.3 常见二叉树类型
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1. 完美二叉树
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2. 完全二叉树
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3. 完满二叉树
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4. 平衡二叉树
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7.1.4 二叉树的退化
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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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<h1 id="71">7.1 二叉树<a class="headerlink" href="#71" title="Permanent link">¶</a></h1>
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<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：值、左子节点引用、右子节点引用。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><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" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" 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">JS</label><label for="__tabbed_1_6">TS</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><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
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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>
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<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>
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<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">// 左子节点引用</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">// 右子节点引用</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 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"># 左子节点引用</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"># 右子节点引用</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><span class="w"> </span><span class="c1">// 左子节点指针</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><span class="w"> </span><span class="c1">// 右子节点指针</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><span class="w"> </span><span class="c1">// 节点值</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">// 左子节点引用</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">// 右子节点引用</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>
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<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">// 左子节点引用</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">// 右子节点引用</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><span class="cm">/* 二叉树节点结构体 */</span>
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<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></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-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></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-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w"> </span><span class="c1">// 节点高度</span>
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<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">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-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">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-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="p">};</span>
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<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a>
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<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">;</span>
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<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a>
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<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="cm">/* 构造函数 */</span>
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<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">newTreeNode</span><span class="p">(</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="p">{</span>
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<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">;</span>
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<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a>
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<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">TreeNode</span><span class="p">));</span>
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<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w"> </span><span class="n">node</span><span class="o">-></span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
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<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w"> </span><span class="n">node</span><span class="o">-></span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
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<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w"> </span><span class="n">node</span><span class="o">-></span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
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<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w"> </span><span class="n">node</span><span class="o">-></span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
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<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
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<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></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-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">// 左子节点引用</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">// 右子节点引用</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">// 左子节点引用</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">// 右子节点引用</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 class="tabbed-block">
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<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 二叉树节点类 */</span>
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<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
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|
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<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></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-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></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">// 左子节点引用</span>
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<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></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">// 右子节点引用</span>
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<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">right</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="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-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></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>每个节点都有两个引用（指针），分别指向「左子节点 left-child node」和「右子节点 right-child node」，该节点被称为这两个子节点的「父节点 parent node」。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 left subtree」，同理可得「右子树 right subtree」。</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>。如图 7-1 所示，如果将“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"> 图 7-1 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>二叉树的常用术语如图 7-2 所示。</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">\(\text{None}\)</span> 。</li>
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|
<li>「边 edge」：连接两个节点的线段，即节点引用（指针）。</li>
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|
<li>节点所在的「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，度的取值范围是 0、1、2 。</li>
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|
<li>二叉树的「高度 height」：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"> 图 7-2 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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<h3 id="1">1. 初始化二叉树<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
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<p>与链表类似，首先初始化节点，然后构建引用（指针）。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><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">JS</label><label for="__tabbed_2_6">TS</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><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
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<div class="tabbed-content">
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<div class="highlight"><span class="filename">binary_tree.java</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></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-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></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-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></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-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></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-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></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-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></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-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></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-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></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-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></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-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="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-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></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-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></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-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></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-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></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-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="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-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></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-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></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-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></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-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="c1"># 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="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-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></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-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></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-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></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-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></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-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="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-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></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-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></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-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></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-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="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-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></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-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></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-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></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-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></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-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="p">=</span><span class="w"> </span><span class="nx">n2</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="p">=</span><span class="w"> </span><span class="nx">n3</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="p">=</span><span class="w"> </span><span class="nx">n4</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="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-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></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-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></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-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></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-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></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-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></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-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></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-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></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-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></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-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></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-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="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-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></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-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></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-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></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-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></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-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="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-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></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-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></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-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></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-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">1</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">2</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">3</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">4</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="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
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<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="c1">// 构建引用指向（即指针）</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="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-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></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-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></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-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></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.cs</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></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-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></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-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></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-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></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-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></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-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></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-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></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-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></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-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></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-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></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-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></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-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></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-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></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-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></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-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></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-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></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-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></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-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></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-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></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.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 初始化二叉树 */</span>
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<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// 初始化节点</span>
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<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></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-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></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-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></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-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></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-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></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-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="c1">// 构建引用指向（即指针）</span>
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<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></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-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></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-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></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-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></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.rs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></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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<h3 id="2">2. 插入与删除节点<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
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<p>与链表类似，在二叉树中插入与删除节点可以通过修改指针来实现。图 7-3 给出了一个示例。</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"> 图 7-3 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="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><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">JS</label><label for="__tabbed_3_6">TS</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><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</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-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></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-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></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-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></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-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></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-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></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-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>
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<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></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-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></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-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></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>
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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-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="c1"># 插入与删除节点</span>
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<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></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>
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<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="c1"># 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></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-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></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-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="c1"># 删除节点 P</span>
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<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></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-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></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>
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<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></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>
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<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></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>
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<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></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>
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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-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></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>
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<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></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>
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<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></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>
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<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></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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</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-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></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>
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<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></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>
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<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></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>
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<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></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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</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-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
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<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></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-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></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-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></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>
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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-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></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>
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<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></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>
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<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></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>
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<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></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>
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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-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></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>
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<a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">P</span>
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<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="n">P</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n2</span>
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<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span> <span class="p">=</span> <span class="n">n2</span>
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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-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></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.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 插入与删除节点 */</span>
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<a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></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>
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<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="c1">// 在 n1 -> n2 中间插入节点 P</span>
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<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></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>
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<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></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-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="c1">// 删除节点 P</span>
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<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></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>
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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.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></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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<div class="admonition note">
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|
<p class="admonition-title">Note</p>
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|
<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>
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</div>
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<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_1">1. 完美二叉树<a class="headerlink" href="#1_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>请注意，在中文社区中，完美二叉树常被称为「满二叉树」。</p>
|
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</div>
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|
|
<p><img alt="完美二叉树" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
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<p align="center"> 图 7-4 完美二叉树 </p>
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<h3 id="2_1">2. 完全二叉树<a class="headerlink" href="#2_1" title="Permanent link">¶</a></h3>
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|
<p>如图 7-5 所示，「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>
|
|
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<p><img alt="完全二叉树" src="../binary_tree.assets/complete_binary_tree.png" /></p>
|
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<p align="center"> 图 7-5 完全二叉树 </p>
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<h3 id="3">3. 完满二叉树<a class="headerlink" href="#3" title="Permanent link">¶</a></h3>
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<p>如图 7-6 所示，「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>
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|
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<p><img alt="完满二叉树" src="../binary_tree.assets/full_binary_tree.png" /></p>
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<p align="center"> 图 7-6 完满二叉树 </p>
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<h3 id="4">4. 平衡二叉树<a class="headerlink" href="#4" title="Permanent link">¶</a></h3>
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<p>如图 7-7 所示，「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>
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<p><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
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<p align="center"> 图 7-7 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>
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<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>
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<ul>
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<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>
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<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>
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</ul>
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<p><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></p>
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<p align="center"> 图 7-8 二叉树的最佳与最差结构 </p>
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<p>如表 7-1 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>
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<p align="center"> 表 7-1 二叉树的最佳与最差情况 </p>
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<div class="center-table">
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<table>
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<thead>
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<tr>
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<th></th>
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<th>完美二叉树</th>
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<th>链表</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>第 <span class="arithmatex">\(i\)</span> 层的节点数量</td>
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<td><span class="arithmatex">\(2^{i-1}\)</span></td>
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<td><span class="arithmatex">\(1\)</span></td>
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</tr>
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<tr>
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<td>高度 <span class="arithmatex">\(h\)</span> 树的叶节点数量</td>
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<td><span class="arithmatex">\(2^h\)</span></td>
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<td><span class="arithmatex">\(1\)</span></td>
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</tr>
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<tr>
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<td>高度 <span class="arithmatex">\(h\)</span> 树的节点总数</td>
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<td><span class="arithmatex">\(2^{h+1} - 1\)</span></td>
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<td><span class="arithmatex">\(h + 1\)</span></td>
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</tr>
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<tr>
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<td>节点总数 <span class="arithmatex">\(n\)</span> 树的高度</td>
|
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|
<td><span class="arithmatex">\(\log_2 (n+1) - 1\)</span></td>
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|
<td><span class="arithmatex">\(n - 1\)</span></td>
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</tr>
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</tbody>
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</table>
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