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</div>
<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>
<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>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="父节点、子节点、子树" src="../binary_tree.assets/binary_tree_definition.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="父节点、子节点、子树" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></a></p>
<p align="center"> 图 7-1 &nbsp; 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>
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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>
<li>节点的「高度 height」：从距离该节点最远的叶节点到该节点所经过的边的数量。</li>
</ul>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_terminology.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的常用术语" src="../binary_tree.assets/binary_tree_terminology.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_terminology.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的常用术语" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></a></p>
<p align="center"> 图 7-2 &nbsp; 二叉树的常用术语 </p>
<div class="admonition tip">
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</div>
<h3 id="2">2. &nbsp; 插入与删除节点<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<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>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_add_remove.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉树中插入与删除节点" src="../binary_tree.assets/binary_tree_add_remove.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_add_remove.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉树中插入与删除节点" class="animation-figure" src="../binary_tree.assets/binary_tree_add_remove.png" /></a></p>
<p align="center"> 图 7-3 &nbsp; 在二叉树中插入与删除节点 </p>
<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">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Zig</label></div>
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<p class="admonition-title">Tip</p>
<p>请注意，在中文社区中，完美二叉树常被称为「满二叉树」。</p>
</div>
<p><a class="glightbox" href="../binary_tree.assets/perfect_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树" src="../binary_tree.assets/perfect_binary_tree.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/perfect_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></a></p>
<p align="center"> 图 7-4 &nbsp; 完美二叉树 </p>
<h3 id="2_1">2. &nbsp; 完全二叉树<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
<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>
<p><a class="glightbox" href="../binary_tree.assets/complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树" src="../binary_tree.assets/complete_binary_tree.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></a></p>
<p align="center"> 图 7-5 &nbsp; 完全二叉树 </p>
<h3 id="3">3. &nbsp; 完满二叉树<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<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>
<p><a class="glightbox" href="../binary_tree.assets/full_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完满二叉树" src="../binary_tree.assets/full_binary_tree.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/full_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完满二叉树" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></a></p>
<p align="center"> 图 7-6 &nbsp; 完满二叉树 </p>
<h3 id="4">4. &nbsp; 平衡二叉树<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
<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>
<p><a class="glightbox" href="../binary_tree.assets/balanced_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="平衡二叉树" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/balanced_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="平衡二叉树" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
<p align="center"> 图 7-7 &nbsp; 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>
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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>
<li>链表则是另一个极端，各项操作都变为线性操作，时间复杂度退化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
</ul>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_best_worst_cases.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的最佳与最差结构" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></a></p>
<p><a class="glightbox" href="../binary_tree.assets/binary_tree_best_worst_cases.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的最佳与最差结构" class="animation-figure" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></a></p>
<p align="center"> 图 7-8 &nbsp; 二叉树的最佳与最差结构 </p>
<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>