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<title>8.3.   二叉树数组表示 - Hello 算法</title>
<title>7.3.   二叉树数组表示 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
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8.3. &nbsp; 二叉树数组表示
7.3. &nbsp; 二叉树数组表示
</span>
</div>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. &nbsp; &nbsp; 二分查找
6. &nbsp; &nbsp; 散列表
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6. &nbsp; &nbsp; 二分查找
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<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>
<li class="md-nav__item">
<a href="../../chapter_binary_search/binary_search_edge/" class="md-nav__link">
6.2. &nbsp; 二分查找边界
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7. &nbsp; &nbsp; 散列表
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7. &nbsp; &nbsp; 散列表
6. &nbsp; &nbsp; 散列表
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
6.1. &nbsp; 哈希表
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
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7.3. &nbsp; 小结
6.3. &nbsp; 小结
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 树
<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
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8. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
7.1. &nbsp; 二叉树
</a>
</li>
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<li class="md-nav__item">
<a href="../binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
7.2. &nbsp; 二叉树遍历
</a>
</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
8.3. &nbsp; 二叉树数组表示
7.3. &nbsp; 二叉树数组表示
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
8.3. &nbsp; 二叉树数组表示
7.3. &nbsp; 二叉树数组表示
</a>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#831" class="md-nav__link">
8.3.1. &nbsp; 表示完美二叉树
<a href="#731" class="md-nav__link">
7.3.1. &nbsp; 表示完美二叉树
</a>
</li>
<li class="md-nav__item">
<a href="#832" class="md-nav__link">
8.3.2. &nbsp; 表示任意二叉树
<a href="#732" class="md-nav__link">
7.3.2. &nbsp; 表示任意二叉树
</a>
</li>
<li class="md-nav__item">
<a href="#833" class="md-nav__link">
8.3.3. &nbsp; 优势与局限性
<a href="#733" class="md-nav__link">
7.3.3. &nbsp; 优势与局限性
</a>
</li>
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<li class="md-nav__item">
<a href="../binary_search_tree/" class="md-nav__link">
8.4. &nbsp; 二叉搜索树
7.4. &nbsp; 二叉搜索树
</a>
</li>
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<li class="md-nav__item">
<a href="../avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
7.5. &nbsp; AVL 树 *
</a>
</li>
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
8.6. &nbsp; 小结
7.6. &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" >
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8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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8. &nbsp; &nbsp; 堆
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<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
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 堆
9. &nbsp; &nbsp; 图
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9. &nbsp; &nbsp; 堆
9. &nbsp; &nbsp; 图
</label>
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. &nbsp; 图
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. &nbsp; 图基础操作
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
<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>
</li>
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
10.1. &nbsp; 图
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.1. &nbsp; 二分查找（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
10.2. &nbsp; 二分查找边界（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
10.3. &nbsp; 哈希优化策略
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/summary/" class="md-nav__link">
10.4. &nbsp; 小结
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
10.4. &nbsp; 重识搜索算法
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.5. &nbsp; 小结
</a>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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12. &nbsp; &nbsp; 搜索算法
12. &nbsp; &nbsp; 回溯算法
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<li class="md-nav__item">
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12.1. &nbsp; 搜索算法（New）
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
12.1. &nbsp; 回溯算法（New）
</a>
</li>
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<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 href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
12.2. &nbsp; 全排列问题（New）
</a>
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<li class="md-nav__item">
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12.3. &nbsp; 小结
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
12.3. &nbsp; N 皇后问题（New）
</a>
</li>
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13. &nbsp; &nbsp; 回溯算法
13. &nbsp; &nbsp; 附录
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13. &nbsp; &nbsp; 附录
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<li class="md-nav__item">
<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
13.1. &nbsp; 回溯算法（New）
<a href="../../chapter_appendix/installation/" class="md-nav__link">
13.1. &nbsp; 编程环境安装
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
13.2. &nbsp; 全排列问题（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
13.3. &nbsp; N 皇后问题（New）
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
13.2. &nbsp; 一起参与创作
</a>
</li>
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14. &nbsp; &nbsp; 附录
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14. &nbsp; &nbsp; 附录
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<li class="md-nav__item">
<a href="../../chapter_appendix/installation/" class="md-nav__link">
14.1. &nbsp; 编程环境安装
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<li class="md-nav__item">
<a href="../../chapter_appendix/contribution/" class="md-nav__link">
14.2. &nbsp; 一起参与创作
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参考文献
</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#831" class="md-nav__link">
8.3.1. &nbsp; 表示完美二叉树
<a href="#731" class="md-nav__link">
7.3.1. &nbsp; 表示完美二叉树
</a>
</li>
<li class="md-nav__item">
<a href="#832" class="md-nav__link">
8.3.2. &nbsp; 表示任意二叉树
<a href="#732" class="md-nav__link">
7.3.2. &nbsp; 表示任意二叉树
</a>
</li>
<li class="md-nav__item">
<a href="#833" class="md-nav__link">
8.3.3. &nbsp; 优势与局限性
<a href="#733" class="md-nav__link">
7.3.3. &nbsp; 优势与局限性
</a>
</li>
@@ -1927,17 +1888,17 @@
<h1 id="83">8.3. &nbsp; 二叉树数组表示<a class="headerlink" href="#83" title="Permanent link">&para;</a></h1>
<h1 id="73">7.3. &nbsp; 二叉树数组表示<a class="headerlink" href="#73" title="Permanent link">&para;</a></h1>
<p>在链表表示下，二叉树的存储单元为节点 <code>TreeNode</code> ，节点之间通过指针相连接。在上节中，我们学习了在链表表示下的二叉树的各项基本操作。</p>
<p>那么，能否用「数组」来表示二叉树呢？答案是肯定的。</p>
<h2 id="831">8.3.1. &nbsp; 表示完美二叉树<a class="headerlink" href="#831" title="Permanent link">&para;</a></h2>
<h2 id="731">7.3.1. &nbsp; 表示完美二叉树<a class="headerlink" href="#731" title="Permanent link">&para;</a></h2>
<p>先分析一个简单案例，给定一个完美二叉树，我们将节点按照层序遍历的顺序编号（从 <span class="arithmatex">\(0\)</span> 开始），此时每个节点都对应唯一的索引。</p>
<p>根据层序遍历的特性，我们可以推导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><img alt="完美二叉树的数组表示" src="../binary_tree.assets/array_representation_mapping.png" /></p>
<p align="center"> Fig. 完美二叉树的数组表示 </p>
<p><strong>映射公式的作用相当于链表中的指针</strong>。如果我们将节点按照层序遍历的顺序存储在一个数组中，那么对于数组中的任意节点，我们都可以通过映射公式来访问其子节点。</p>
<h2 id="832">8.3.2. &nbsp; 表示任意二叉树<a class="headerlink" href="#832" title="Permanent link">&para;</a></h2>
<h2 id="732">7.3.2. &nbsp; 表示任意二叉树<a class="headerlink" href="#732" title="Permanent link">&para;</a></h2>
<p>然而，完美二叉树只是一个特例。在二叉树的中间层，通常存在许多 <span class="arithmatex">\(\text{null}\)</span> ，而层序遍历序列并不包含这些 <span class="arithmatex">\(\text{null}\)</span> 。我们无法仅凭该序列来推测 <span class="arithmatex">\(\text{null}\)</span> 的数量和分布位置，<strong>这意味着存在多种二叉树结构都符合该层序遍历序列</strong>。显然在这种情况下，上述的数组表示方法已经失效。</p>
<p><img alt="层序遍历序列对应多种二叉树可能性" src="../binary_tree.assets/array_representation_without_empty.png" /></p>
<p align="center"> Fig. 层序遍历序列对应多种二叉树可能性 </p>
@@ -2008,7 +1969,7 @@
<p><img alt="任意类型二叉树的数组表示" src="../binary_tree.assets/array_representation_with_empty.png" /></p>
<p align="center"> Fig. 任意类型二叉树的数组表示 </p>
<h2 id="833">8.3.3. &nbsp; 优势与局限性<a class="headerlink" href="#833" title="Permanent link">&para;</a></h2>
<h2 id="733">7.3.3. &nbsp; 优势与局限性<a class="headerlink" href="#733" title="Permanent link">&para;</a></h2>
<p>二叉树的数组表示存在以下优点：</p>
<ul>
<li>数组存储在连续的内存空间中，缓存友好，访问与遍历速度较快；</li>
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<nav class="md-footer__inner md-grid" aria-label="页脚" >
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