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<li class="md-nav__item">
<a href="#733" class="md-nav__link">
7.3.3 &nbsp; 优势与局限性
7.3.3 &nbsp; 优点与局限性
</a>
</li>
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<li class="md-nav__item">
<a href="#733" class="md-nav__link">
7.3.3 &nbsp; 优势与局限性
7.3.3 &nbsp; 优点与局限性
</a>
</li>
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<!-- Page content -->
<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>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。上一节介绍了链表表示下的二叉树的各项基本操作。</p>
<p>那么，我们能否用数组来表示二叉树呢？答案是肯定的。</p>
<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，我们将所有节点按照层序遍历的顺序存储在一个数组中，则每个节点都对应唯一的数组索引。</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> 。图 7-12 展示了各个节点索引之间的映射关系。</p>
<p>先分析一个简单案例。给定一棵完美二叉树，我们将所有节点按照层序遍历的顺序存储在一个数组中，则每个节点都对应唯一的数组索引。</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>若某节点的索引为 <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> 。图 7-12 展示了各个节点索引之间的映射关系。</p>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树的数组表示" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_binary_tree.png" /></a></p>
<p align="center"> 图 7-12 &nbsp; 完美二叉树的数组表示 </p>
<p><strong>映射公式的角色相当于链表中的指针</strong>。给定数组中的任意一个节点，我们都可以通过映射公式来访问它的左（右）子节点。</p>
<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>完美二叉树是一个特例，在二叉树的中间层通常存在许多 <span class="arithmatex">\(\text{None}\)</span> 。由于层序遍历序列并不包含这些 <span class="arithmatex">\(\text{None}\)</span> ，因此我们无法仅凭该序列来推测 <span class="arithmatex">\(\text{None}\)</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>。</p>
<p>如图 7-13 所示，给定一个非完美二叉树，上述的数组表示方法已经失效。</p>
<p>如图 7-13 所示，给定一棵非完美二叉树，上述数组表示方法已经失效。</p>
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_without_empty.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="层序遍历序列对应多种二叉树可能性" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></a></p>
<p align="center"> 图 7-13 &nbsp; 层序遍历序列对应多种二叉树可能性 </p>
<p>为了解决此问题，<strong>我们可以考虑在层序遍历序列中显式地写出所有 <span class="arithmatex">\(\text{None}\)</span></strong> 。如图 7-14 所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。</p>
<p>为了解决此问题，<strong>我们可以考虑在层序遍历序列中显式地写出所有 <span class="arithmatex">\(\text{None}\)</span></strong> 。如图 7-14 所示，这样处理后，层序遍历序列就可以唯一表示二叉树了。示例代码如下：</p>
<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">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
@@ -3494,7 +3494,7 @@
<p><a class="glightbox" href="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树的数组表示" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></a></p>
<p align="center"> 图 7-15 &nbsp; 完全二叉树的数组表示 </p>
<p>以下代码实现了一个基于数组表示的二叉树，包括以下几种操作。</p>
<p>以下代码实现了一棵基于数组表示的二叉树，包括以下几种操作。</p>
<ul>
<li>给定某节点，获取它的值、左（右）子节点、父节点。</li>
<li>获取前序遍历、中序遍历、后序遍历、层序遍历序列。</li>
@@ -4071,7 +4071,7 @@
<a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a>
<a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的父节点的索引 */</span>
<a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a><span class="w"> </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// 向下取整</span>
<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// 向下整除</span>
<a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a>
<a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w"> </span><span class="cm">/* 层序遍历 */</span>
@@ -4155,7 +4155,7 @@
<a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a>
<a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w"> </span><span class="cm">/* 获取索引为 i 节点的父节点的索引 */</span>
<a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w"> </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// 向下取整</span>
<a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// 向下整除</span>
<a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a>
<a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w"> </span><span class="cm">/* 层序遍历 */</span>
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</div>
</div>
</div>
<h2 id="733">7.3.3 &nbsp; 优势与局限性<a class="headerlink" href="#733" title="Permanent link">&para;</a></h2>
<h2 id="733">7.3.3 &nbsp; 优点与局限性<a class="headerlink" href="#733" title="Permanent link">&para;</a></h2>
<p>二叉树的数组表示主要有以下优点。</p>
<ul>
<li>数组存储在连续的内存空间中，对缓存友好，访问与遍历速度较快。</li>