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<title>7.1.   二叉树 - Hello 算法</title>
<title>8.1.   二叉树 - Hello 算法</title>
@ -79,7 +79,7 @@
<div data-md-component="skip">
<a href="#71" class="md-skip">
<a href="#81" class="md-skip">
跳转至
</a>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
</span>
</div>
@ -441,8 +441,6 @@
<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
@ -504,23 +502,9 @@
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. &nbsp; 权衡时间与空间
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
</a>
</li>
@ -847,21 +831,76 @@
<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
6. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 二分查找
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 散列表
<span class="md-nav__icon md-icon"></span>
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<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
7. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
7.1. &nbsp; 哈希表
</a>
</li>
@ -886,7 +925,7 @@
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
7.2. &nbsp; 哈希冲突处理
</a>
</li>
@ -900,7 +939,7 @@
<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
7.3. &nbsp; 小结
</a>
</li>
@ -930,7 +969,7 @@
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_8">
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
7. &nbsp; &nbsp; 树
8. &nbsp; &nbsp; 树
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -974,12 +1013,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
</a>
@ -998,25 +1037,25 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#711" class="md-nav__link">
7.1.1. &nbsp; 二叉树常见术语
<a href="#811" class="md-nav__link">
8.1.1. &nbsp; 二叉树常见术语
</a>
</li>
<li class="md-nav__item">
<a href="#712" class="md-nav__link">
7.1.2. &nbsp; 二叉树基本操作
<a href="#812" class="md-nav__link">
8.1.2. &nbsp; 二叉树基本操作
</a>
</li>
<li class="md-nav__item">
<a href="#713" class="md-nav__link">
7.1.3. &nbsp; 常见二叉树类型
<a href="#813" class="md-nav__link">
8.1.3. &nbsp; 常见二叉树类型
</a>
<nav class="md-nav" aria-label="7.1.3. &nbsp; 常见二叉树类型">
<nav class="md-nav" aria-label="8.1.3. &nbsp; 常见二叉树类型">
<ul class="md-nav__list">
<li class="md-nav__item">
@ -1053,15 +1092,15 @@
</li>
<li class="md-nav__item">
<a href="#714" class="md-nav__link">
7.1.4. &nbsp; 二叉树的退化
<a href="#814" class="md-nav__link">
8.1.4. &nbsp; 二叉树的退化
</a>
</li>
<li class="md-nav__item">
<a href="#715" class="md-nav__link">
7.1.5. &nbsp; 二叉树表示方式 *
<a href="#815" class="md-nav__link">
8.1.5. &nbsp; 二叉树表示方式 *
</a>
</li>
@ -1082,7 +1121,7 @@
<li class="md-nav__item">
<a href="../binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
8.2. &nbsp; 二叉树遍历
</a>
</li>
@ -1096,7 +1135,7 @@
<li class="md-nav__item">
<a href="../binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
8.3. &nbsp; 二叉搜索树
</a>
</li>
@ -1110,7 +1149,7 @@
<li class="md-nav__item">
<a href="../avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
8.4. &nbsp; AVL 树 *
</a>
</li>
@ -1124,94 +1163,7 @@
<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
7.5. &nbsp; 小结
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
8. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. &nbsp; 堆
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
8.5. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1271,8 +1221,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. &nbsp; 图
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
</a>
</li>
@ -1285,8 +1235,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
</a>
</li>
@ -1299,22 +1249,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
9.4. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
</a>
</li>
@ -1357,14 +1293,14 @@
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 查找算法
10. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
10. &nbsp; &nbsp; 查找算法
10. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1374,8 +1310,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
<a href="../../chapter_graph/graph/" class="md-nav__link">
10.1. &nbsp; 图
</a>
</li>
@ -1388,8 +1324,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. &nbsp; 二分查找
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
</a>
</li>
@ -1402,8 +1338,8 @@
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
</a>
</li>
@ -1416,7 +1352,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
<a href="../../chapter_graph/summary/" class="md-nav__link">
10.4. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
</li>
@ -1572,7 +1508,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
</a>
</li>
@ -1586,7 +1522,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
</li>
@ -1634,17 +1570,21 @@
<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_13">
<span class="md-nav__icon md-icon"></span>
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1654,8 +1594,36 @@
<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法（New）
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
12.2. &nbsp; 哈希优化策略
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
12.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_14">
<span class="md-nav__icon md-icon"></span>
13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@ -1711,22 +1677,8 @@
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
</a>
</li>
@ -1759,6 +1711,77 @@
<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
14. &nbsp; &nbsp; 附录
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_15">
<span class="md-nav__icon md-icon"></span>
14. &nbsp; &nbsp; 附录
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
@ -1771,8 +1794,8 @@
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_15">
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_16">
<span class="md-nav__icon md-icon"></span>
参考文献
</label>
@ -1813,25 +1836,25 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#711" class="md-nav__link">
7.1.1. &nbsp; 二叉树常见术语
<a href="#811" class="md-nav__link">
8.1.1. &nbsp; 二叉树常见术语
</a>
</li>
<li class="md-nav__item">
<a href="#712" class="md-nav__link">
7.1.2. &nbsp; 二叉树基本操作
<a href="#812" class="md-nav__link">
8.1.2. &nbsp; 二叉树基本操作
</a>
</li>
<li class="md-nav__item">
<a href="#713" class="md-nav__link">
7.1.3. &nbsp; 常见二叉树类型
<a href="#813" class="md-nav__link">
8.1.3. &nbsp; 常见二叉树类型
</a>
<nav class="md-nav" aria-label="7.1.3. &nbsp; 常见二叉树类型">
<nav class="md-nav" aria-label="8.1.3. &nbsp; 常见二叉树类型">
<ul class="md-nav__list">
<li class="md-nav__item">
@ -1868,15 +1891,15 @@
</li>
<li class="md-nav__item">
<a href="#714" class="md-nav__link">
7.1.4. &nbsp; 二叉树的退化
<a href="#814" class="md-nav__link">
8.1.4. &nbsp; 二叉树的退化
</a>
</li>
<li class="md-nav__item">
<a href="#715" class="md-nav__link">
7.1.5. &nbsp; 二叉树表示方式 *
<a href="#815" class="md-nav__link">
8.1.5. &nbsp; 二叉树表示方式 *
</a>
</li>
@ -1904,7 +1927,7 @@
<h1 id="71">7.1. &nbsp; 二叉树<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
<h1 id="81">8.1. &nbsp; 二叉树<a class="headerlink" href="#81" title="Permanent link">&para;</a></h1>
<p>「二叉树 Binary Tree」是一种非线性数据结构，代表着祖先与后代之间的派生关系，体现着“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含一个「值」和两个「指针」。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label></div>
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@ -2016,7 +2039,7 @@
<p><img alt="父节点、子节点、子树" src="../binary_tree.assets/binary_tree_definition.png" /></p>
<p align="center"> Fig. 父节点、子节点、子树 </p>
<h2 id="711">7.1.1. &nbsp; 二叉树常见术语<a class="headerlink" href="#711" title="Permanent link">&para;</a></h2>
<h2 id="811">8.1.1. &nbsp; 二叉树常见术语<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
<p>二叉树涉及的术语较多，建议尽量理解并记住。</p>
<ul>
<li>「根节点 Root Node」：位于二叉树顶层的节点，没有父节点；</li>
@ -2035,7 +2058,7 @@
<p class="admonition-title">高度与深度的定义</p>
<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>
</div>
<h2 id="712">7.1.2. &nbsp; 二叉树基本操作<a class="headerlink" href="#712" title="Permanent link">&para;</a></h2>
<h2 id="812">8.1.2. &nbsp; 二叉树基本操作<a class="headerlink" href="#812" title="Permanent link">&para;</a></h2>
<p><strong>初始化二叉树</strong>。与链表类似，首先初始化节点，然后构建引用指向（即指针）。</p>
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<p class="admonition-title">Note</p>
<p>需要注意的是，插入节点可能会改变二叉树的原有逻辑结构，而删除节点通常意味着删除该节点及其所有子树。因此，在二叉树中，插入与删除操作通常是由一套操作配合完成的，以实现有实际意义的操作。</p>
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<h2 id="713">7.1.3. &nbsp; 常见二叉树类型<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
<h2 id="813">8.1.3. &nbsp; 常见二叉树类型<a class="headerlink" href="#813" title="Permanent link">&para;</a></h2>
<h3 id="_1">完美二叉树<a class="headerlink" href="#_1" title="Permanent link">&para;</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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<p><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
<p align="center"> Fig. 平衡二叉树 </p>
<h2 id="714">7.1.4. &nbsp; 二叉树的退化<a class="headerlink" href="#714" title="Permanent link">&para;</a></h2>
<h2 id="814">8.1.4. &nbsp; 二叉树的退化<a class="headerlink" href="#814" title="Permanent link">&para;</a></h2>
<p>当二叉树的每层节点都被填满时，达到「完美二叉树」；而当所有节点都偏向一侧时，二叉树退化为「链表」。</p>
<ul>
<li>完美二叉树是理想情况，可以充分发挥二叉树“分治”的优势；</li>
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<h2 id="715">7.1.5. &nbsp; 二叉树表示方式 *<a class="headerlink" href="#715" title="Permanent link">&para;</a></h2>
<h2 id="815">8.1.5. &nbsp; 二叉树表示方式 *<a class="headerlink" href="#815" title="Permanent link">&para;</a></h2>
<p>我们通常使用二叉树的「链表表示」，即存储单位为节点 <code>TreeNode</code> ，节点之间通过指针相连接。本文前述示例代码展示了二叉树在链表表示下的各项基本操作。</p>
<p>那么，能否用「数组」来表示二叉树呢？答案是肯定的。先来分析一个简单案例，给定一个「完美二叉树」，将节点按照层序遍历的顺序编号（从 0 开始），那么可以推导得出父节点索引与子节点索引之间的“映射公式”：<strong>若节点的索引为 <span class="arithmatex">\(i\)</span> ，则该节点的左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span></strong> 。</p>
<p><strong>本质上，映射公式的作用相当于链表中的指针</strong>。对于层序遍历序列中的任意节点，我们都可以使用映射公式来访问其子节点。因此，我们可以将二叉树的层序遍历序列存储到数组中，利用以上映射公式来表示二叉树。</p>
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