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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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14.7 &nbsp; 小结
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<li>对于根节点，左子树中所有节点的值 <span class="arithmatex">\(&lt;\)</span> 根节点的值 <span class="arithmatex">\(&lt;\)</span> 右子树中所有节点的值。</li>
<li>任意节点的左、右子树也是二叉搜索树，即同样满足条件 <code>1.</code> 。</li>
</ol>
<p><img alt="二叉搜索树" src="../binary_search_tree.assets/binary_search_tree.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/binary_search_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树" src="../binary_search_tree.assets/binary_search_tree.png" /></a></p>
<p align="center"> 图 7-16 &nbsp; 二叉搜索树 </p>
<h2 id="741">7.4.1 &nbsp; 二叉搜索树的操作<a class="headerlink" href="#741" title="Permanent link">&para;</a></h2>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:4"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label></div>
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<p><img alt="二叉搜索树查找节点示例" src="../binary_search_tree.assets/bst_search_step1.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树查找节点示例" src="../binary_search_tree.assets/bst_search_step1.png" /></a></p>
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<p><img alt="bst_search_step2" src="../binary_search_tree.assets/bst_search_step2.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_search_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="bst_search_step2" src="../binary_search_tree.assets/bst_search_step2.png" /></a></p>
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<p><img alt="bst_search_step3" src="../binary_search_tree.assets/bst_search_step3.png" /></p>
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<p><img alt="bst_search_step4" src="../binary_search_tree.assets/bst_search_step4.png" /></p>
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<li><strong>查找插入位置</strong>：与查找操作相似，从根节点出发，根据当前节点值和 <code>num</code> 的大小关系循环向下搜索，直到越过叶节点（遍历至 <span class="arithmatex">\(\text{None}\)</span> ）时跳出循环。</li>
<li><strong>在该位置插入节点</strong>：初始化节点 <code>num</code> ，将该节点置于 <span class="arithmatex">\(\text{None}\)</span> 的位置。</li>
</ol>
<p><img alt="在二叉搜索树中插入节点" src="../binary_search_tree.assets/bst_insert.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_insert.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中插入节点" src="../binary_search_tree.assets/bst_insert.png" /></a></p>
<p align="center"> 图 7-18 &nbsp; 在二叉搜索树中插入节点 </p>
<p>在代码实现中，需要注意以下两点。</p>
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<p>与插入节点类似，我们需要保证在删除操作完成后，二叉搜索树的“左子树 &lt; 根节点 &lt; 右子树”的性质仍然满足。</p>
<p>因此，我们需要根据目标节点的子节点数量，共分为 0、1 和 2 这三种情况，执行对应的删除节点操作。</p>
<p>如图 7-19 所示，当待删除节点的度为 <span class="arithmatex">\(0\)</span> 时，表示该节点是叶节点，可以直接删除。</p>
<p><img alt="在二叉搜索树中删除节点（度为 0 ）" src="../binary_search_tree.assets/bst_remove_case1.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中删除节点（度为 0 ）" src="../binary_search_tree.assets/bst_remove_case1.png" /></a></p>
<p align="center"> 图 7-19 &nbsp; 在二叉搜索树中删除节点（度为 0 ） </p>
<p>如图 7-20 所示，当待删除节点的度为 <span class="arithmatex">\(1\)</span> 时，将待删除节点替换为其子节点即可。</p>
<p><img alt="在二叉搜索树中删除节点（度为 1 ）" src="../binary_search_tree.assets/bst_remove_case2.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中删除节点（度为 1 ）" src="../binary_search_tree.assets/bst_remove_case2.png" /></a></p>
<p align="center"> 图 7-20 &nbsp; 在二叉搜索树中删除节点（度为 1 ） </p>
<p>当待删除节点的度为 <span class="arithmatex">\(2\)</span> 时，我们无法直接删除它，而需要使用一个节点替换该节点。由于要保持二叉搜索树“左 <span class="arithmatex">\(&lt;\)</span> 根 <span class="arithmatex">\(&lt;\)</span> 右”的性质，<strong>因此这个节点可以是右子树的最小节点或左子树的最大节点</strong>。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:4"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label></div>
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<p><img alt="在二叉搜索树中删除节点（度为 2 ）" src="../binary_search_tree.assets/bst_remove_case3_step1.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_remove_case3_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在二叉搜索树中删除节点（度为 2 ）" src="../binary_search_tree.assets/bst_remove_case3_step1.png" /></a></p>
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<p><img alt="bst_remove_case3_step3" src="../binary_search_tree.assets/bst_remove_case3_step3.png" /></p>
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<p><img alt="bst_remove_case3_step4" src="../binary_search_tree.assets/bst_remove_case3_step4.png" /></p>
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<p>如图 7-22 所示，二叉树的中序遍历遵循“左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右”的遍历顺序，而二叉搜索树满足“左子节点 <span class="arithmatex">\(&lt;\)</span> 根节点 <span class="arithmatex">\(&lt;\)</span> 右子节点”的大小关系。</p>
<p>这意味着在二叉搜索树中进行中序遍历时，总是会优先遍历下一个最小节点，从而得出一个重要性质：<strong>二叉搜索树的中序遍历序列是升序的</strong>。</p>
<p>利用中序遍历升序的性质，我们在二叉搜索树中获取有序数据仅需 <span class="arithmatex">\(O(n)\)</span> 时间，无须进行额外的排序操作，非常高效。</p>
<p><img alt="二叉搜索树的中序遍历序列" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_inorder_traversal.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的中序遍历序列" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></a></p>
<p align="center"> 图 7-22 &nbsp; 二叉搜索树的中序遍历序列 </p>
<h2 id="742">7.4.2 &nbsp; 二叉搜索树的效率<a class="headerlink" href="#742" title="Permanent link">&para;</a></h2>
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<p>在理想情况下，二叉搜索树是“平衡”的，这样就可以在 <span class="arithmatex">\(\log n\)</span> 轮循环内查找任意节点。</p>
<p>然而，如果我们在二叉搜索树中不断地插入和删除节点，可能导致二叉树退化为图 7-23 所示的链表，这时各种操作的时间复杂度也会退化为 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p><img alt="二叉搜索树的退化" src="../binary_search_tree.assets/bst_degradation.png" /></p>
<p><a class="glightbox" href="../binary_search_tree.assets/bst_degradation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的退化" src="../binary_search_tree.assets/bst_degradation.png" /></a></p>
<p align="center"> 图 7-23 &nbsp; 二叉搜索树的退化 </p>
<h2 id="743">7.4.3 &nbsp; 二叉搜索树常见应用<a class="headerlink" href="#743" title="Permanent link">&para;</a></h2>
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