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<title>7.2.   二叉树遍历 - Hello 算法</title>
<title>8.2.   二叉树遍历 - Hello 算法</title>
@@ -79,7 +79,7 @@
<div data-md-component="skip">
<a href="#72" class="md-skip">
<a href="#82" class="md-skip">
跳转至
</a>
@@ -113,7 +113,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
7.2. &nbsp; 二叉树遍历
8.2. &nbsp; 二叉树遍历
</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>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
<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>
@@ -872,7 +911,7 @@
<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>
@@ -946,15 +985,15 @@
<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>
@@ -965,7 +1004,7 @@
<li class="md-nav__item">
<a href="../binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
</a>
</li>
@@ -988,12 +1027,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
7.2. &nbsp; 二叉树遍历
8.2. &nbsp; 二叉树遍历
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
7.2. &nbsp; 二叉树遍历
8.2. &nbsp; 二叉树遍历
</a>
@@ -1012,11 +1051,11 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#721" class="md-nav__link">
7.2.1. &nbsp; 层序遍历
<a href="#821" class="md-nav__link">
8.2.1. &nbsp; 层序遍历
</a>
<nav class="md-nav" aria-label="7.2.1. &nbsp; 层序遍历">
<nav class="md-nav" aria-label="8.2.1. &nbsp; 层序遍历">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -1039,11 +1078,11 @@
</li>
<li class="md-nav__item">
<a href="#722" class="md-nav__link">
7.2.2. &nbsp; 前序、中序、后序遍历
<a href="#822" class="md-nav__link">
8.2.2. &nbsp; 前序、中序、后序遍历
</a>
<nav class="md-nav" aria-label="7.2.2. &nbsp; 前序、中序、后序遍历">
<nav class="md-nav" aria-label="8.2.2. &nbsp; 前序、中序、后序遍历">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -1081,7 +1120,7 @@
<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>
@@ -1095,7 +1134,7 @@
<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>
@@ -1109,94 +1148,7 @@
<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>
@@ -1234,19 +1186,17 @@
<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>
@@ -1256,8 +1206,8 @@
<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>
@@ -1270,8 +1220,8 @@
<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>
@@ -1284,22 +1234,8 @@
<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>
@@ -1342,14 +1278,14 @@
<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>
@@ -1359,8 +1295,8 @@
<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>
@@ -1373,8 +1309,8 @@
<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>
@@ -1387,8 +1323,8 @@
<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>
@@ -1401,7 +1337,7 @@
<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>
@@ -1543,7 +1479,7 @@
<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>
@@ -1557,7 +1493,7 @@
<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>
@@ -1571,7 +1507,7 @@
<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>
@@ -1619,17 +1555,21 @@
<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>
@@ -1639,8 +1579,36 @@
<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>
@@ -1674,19 +1642,17 @@
<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">
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13. &nbsp; &nbsp; 附录
13. &nbsp; &nbsp; 回溯算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1696,22 +1662,8 @@
<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>
@@ -1744,6 +1696,77 @@
<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" >
@@ -1756,8 +1779,8 @@
</div>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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<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>
@@ -1798,11 +1821,11 @@
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#721" class="md-nav__link">
7.2.1. &nbsp; 层序遍历
<a href="#821" class="md-nav__link">
8.2.1. &nbsp; 层序遍历
</a>
<nav class="md-nav" aria-label="7.2.1. &nbsp; 层序遍历">
<nav class="md-nav" aria-label="8.2.1. &nbsp; 层序遍历">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -1825,11 +1848,11 @@
</li>
<li class="md-nav__item">
<a href="#722" class="md-nav__link">
7.2.2. &nbsp; 前序、中序、后序遍历
<a href="#822" class="md-nav__link">
8.2.2. &nbsp; 前序、中序、后序遍历
</a>
<nav class="md-nav" aria-label="7.2.2. &nbsp; 前序、中序、后序遍历">
<nav class="md-nav" aria-label="8.2.2. &nbsp; 前序、中序、后序遍历">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -1874,10 +1897,10 @@
<h1 id="72">7.2. &nbsp; 二叉树遍历<a class="headerlink" href="#72" title="Permanent link">&para;</a></h1>
<h1 id="82">8.2. &nbsp; 二叉树遍历<a class="headerlink" href="#82" title="Permanent link">&para;</a></h1>
<p>从物理结构的角度来看，树是一种基于链表的数据结构，因此其遍历方式是通过指针逐个访问节点。然而，树是一种非线性数据结构，这使得遍历树比遍历链表更加复杂，需要借助搜索算法来实现。</p>
<p>二叉树常见的遍历方式包括层序遍历、前序遍历、中序遍历和后序遍历等。</p>
<h2 id="721">7.2.1. &nbsp; 层序遍历<a class="headerlink" href="#721" title="Permanent link">&para;</a></h2>
<h2 id="821">8.2.1. &nbsp; 层序遍历<a class="headerlink" href="#821" title="Permanent link">&para;</a></h2>
<p>「层序遍历 Level-Order Traversal」从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
<p>层序遍历本质上属于「广度优先搜索 Breadth-First Traversal」，它体现了一种“一圈一圈向外扩展”的逐层搜索方式。</p>
<p><img alt="二叉树的层序遍历" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></p>
@@ -2094,7 +2117,7 @@
<h3 id="_2">复杂度分析<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<p><strong>时间复杂度</strong>：所有节点被访问一次，使用 <span class="arithmatex">\(O(n)\)</span> 时间，其中 <span class="arithmatex">\(n\)</span> 为节点数量。</p>
<p><strong>空间复杂度</strong>：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 <span class="arithmatex">\(\frac{n + 1}{2}\)</span> 个节点，占用 <span class="arithmatex">\(O(n)\)</span> 空间。</p>
<h2 id="722">7.2.2. &nbsp; 前序、中序、后序遍历<a class="headerlink" href="#722" title="Permanent link">&para;</a></h2>
<h2 id="822">8.2.2. &nbsp; 前序、中序、后序遍历<a class="headerlink" href="#822" title="Permanent link">&para;</a></h2>
<p>相应地，前序、中序和后序遍历都属于「深度优先遍历 Depth-First Traversal」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
<p>如下图所示，左侧是深度优先遍历的示意图，右上方是对应的递归实现代码。深度优先遍历就像是绕着整个二叉树的外围“走”一圈，在这个过程中，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。</p>
<p><img alt="二叉搜索树的前、中、后序遍历" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></p>
@@ -2513,7 +2536,7 @@
<nav class="md-footer__inner md-grid" aria-label="页脚" >
<a href="../binary_tree/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.1. &amp;nbsp; 二叉树" rel="prev">
<a href="../binary_tree/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 8.1. &amp;nbsp; 二叉树" rel="prev">
<div class="md-footer__button md-icon">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
</div>
@@ -2522,20 +2545,20 @@
上一页
</span>
<div class="md-ellipsis">
7.1. &nbsp; 二叉树
8.1. &nbsp; 二叉树
</div>
</div>
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