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第 8 章 &nbsp; 堆
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第 8 章 &nbsp; 堆
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8.1 &nbsp; 堆
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8.2 &nbsp; 建堆操作
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8.3 &nbsp; Top-k 问题
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第 9 章 &nbsp; 图
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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9.2 &nbsp; 图基础操作
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9.3 &nbsp; 图的遍历
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9.4 &nbsp; 小结
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第 10 章 &nbsp; 搜索
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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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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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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10.6 &nbsp; 小结
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10.7 &nbsp; 练习
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序算法
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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11.10 &nbsp; 基数排序
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11.11 &nbsp; 小结
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11.12 &nbsp; 练习
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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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12.6 &nbsp; 练习
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第 13 章 &nbsp; 回溯
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13.1 &nbsp; 回溯算法
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13.2 &nbsp; 全排列问题
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13.3 &nbsp; 子集和问题
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13.4 &nbsp; N 皇后问题
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13.5 &nbsp; 小结
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13.6 &nbsp; 练习
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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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<h1 id="77">7.7 &nbsp; 练习<a class="headerlink" href="#77" title="Permanent link">&para;</a></h1>
<h2 id="771">7.7.1 &nbsp; 知识巩固<a class="headerlink" href="#771" title="Permanent link">&para;</a></h2>
<h3 id="1">1. &nbsp; 完全、完满与完美二叉树<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>下面两个数组按层序表示二叉树，<code>None</code> 表示空位：</p>
<ul>
<li>树 A：<code>[1, 2, 3, 4, 5, 6]</code></li>
<li>树 B：<code>[1, 2, 3, None, None, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>哪一棵是完全二叉树？</li>
<li>哪一棵是完满二叉树，即每个非叶节点都有两个子节点？</li>
<li>两棵树中是否有完美二叉树？分别说明理由。</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>树 A 是完全二叉树。它只有最底层未填满，并且节点从左到右连续排列。
树 B 不是完全二叉树，因为最底层左侧已有空位，右侧却仍有节点。</p>
</li>
<li>
<p>树 B 是完满二叉树：节点 1 和节点 3 各有两个子节点，其余节点都是叶节点。
树 A 不是完满二叉树，因为节点 3 只有左子节点 6。</p>
</li>
<li>
<p>两棵都不是完美二叉树，因为它们的最底层都没有完全填满。</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 同一棵树的三种遍历顺序<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>将数组 <code>[1, 2, 3, 4, 5, 6, 7]</code> 按层序存入一棵完全二叉树。</p>
<!-- numbered-subquestions -->
<ol>
<li>画出这棵树。</li>
<li>写出它的前序、中序、后序遍历序列。</li>
<li>在中序序列中，根节点 1 左侧和右侧的两段序列，分别对应树的哪一部分？</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>这棵树为：</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a> 1
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> / \
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> 2 3
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> / \ / \
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> 4 5 6 7
</code></pre></div>
</li>
<li>
<p>前序遍历为 <code>1, 2, 4, 5, 3, 6, 7</code>；
中序遍历为 <code>4, 2, 5, 1, 6, 3, 7</code>；
后序遍历为 <code>4, 5, 2, 6, 7, 3, 1</code>。</p>
</li>
<li>
<p>根节点 1 左侧的 <code>4, 2, 5</code> 是左子树的中序遍历序列；
右侧的 <code>6, 3, 7</code> 是右子树的中序遍历序列。</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 比较两棵二叉搜索树<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>将下面两组序列从左到右依次插入空的二叉搜索树：</p>
<ul>
<li>序列 A：<code>[4, 2, 6, 1, 3, 5, 7]</code></li>
<li>序列 B：<code>[1, 2, 3, 4, 5, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>分别写出查找数字 7 时经过的节点。</li>
<li>若高度按根节点到最远叶节点经过的边数计算，两棵树的高度分别是多少？</li>
<li>根据前两问，你认为两棵树查找数字 7 时的效率是否相同？请结合两棵树的形状和查找路径说明理由。</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>序列 A 建成的树中，查找路径为 <code>4 → 6 → 7</code>；
序列 B 建成的树中，查找路径为 <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>。</p>
</li>
<li>
<p>第一棵树的每一层都填满，高度为 2；第二棵树只有右孩子，高度为 6。</p>
</li>
<li>
<p>两棵树查找 7 时的效率不同。插入顺序改变了二叉搜索树的形状和高度。查找 7 时，第一棵树只经过 3 个节点，
第二棵树则要依次经过 7 个节点；树越高，最坏情况下需要沿路径比较的节点就越多。</p>
</li>
</ol>
</details>
<h2 id="772">7.7.2 &nbsp; 编程练习<a class="headerlink" href="#772" title="Permanent link">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 二叉树的最大深度<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
<p>给定一棵二叉树的根节点 <code>root</code>。每个节点都包含一个整数值，以及指向左、右子节点的引用。</p>
<p>把从根节点到最远叶节点经过的**节点数**称为二叉树的最大深度。请返回这棵树的最大深度；空树的最大深度为 0。
请使用递归完成。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>本题按节点数计算深度：只有一个根节点时，最大深度为 1</li>
<li>让递归函数返回以当前节点为根的子树的最大深度</li>
<li>空节点返回 0，非空节点返回 max(depth(left), depth(right)) + 1</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.cn/problems/maximum-depth-of-binary-tree/" rel="noopener noreferrer" target="_blank">LeetCode</a> <a class="rounded-button exercise-button" href="https://leetcode.cn/problems/maximum-depth-of-binary-tree/solutions/2361697/104-er-cha-shu-de-zui-da-shen-du-hou-xu-txzrx/" rel="noopener noreferrer" target="_blank">题目解析</a></p>
<h3 id="2_1">2. &nbsp; 按层遍历二叉树<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
<p>给定一棵二叉树的根节点 <code>root</code>。请使用队列从上到下逐层访问所有节点，同一层内按照从左到右的顺序访问。</p>
<p>返回一个二维数组：第一个子数组保存根节点所在层的节点值，第二个子数组保存下一层的节点值，依此类推。
若二叉树为空，返回空数组。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>层序遍历需要先进入的节点先访问，因此使用队列</li>
<li>一轮开始时队列中的节点正好属于同一层</li>
<li>先记录队列长度，再弹出这么多个节点并加入它们的孩子</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.cn/problems/binary-tree-level-order-traversal/" rel="noopener noreferrer" target="_blank">LeetCode</a> <a class="rounded-button exercise-button" href="https://leetcode.cn/problems/binary-tree-level-order-traversal/solutions/2361604/102-er-cha-shu-de-ceng-xu-bian-li-yan-du-dyf7/" rel="noopener noreferrer" target="_blank">题目解析</a></p>
<h3 id="3-k">3. &nbsp; 二叉搜索树中的第 k 小元素<a class="headerlink" href="#3-k" title="Permanent link">&para;</a></h3>
<p>一棵二叉搜索树共有 <code>n</code> 个节点，各节点值互不相同。
将所有节点值从小到大排列后，位置从 1 开始编号。</p>
<p>给定根节点 <code>root</code> 和满足 <code>1 &lt;= k &lt;= n</code> 的整数 <code>k</code>，请返回排在第 <code>k</code> 位的节点值。
请在中序遍历过程中直接寻找答案，不先收集全部节点值。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>二叉搜索树的中序遍历会按从小到大的顺序访问节点值</li>
<li>中序遍历依次处理左子树、当前节点和右子树；访问当前节点时把计数加 1</li>
<li>当计数第一次等于 k 时，当前节点值就是答案，此后无须继续遍历</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.cn/problems/kth-smallest-element-in-a-bst/" rel="noopener noreferrer" target="_blank">LeetCode</a> <a class="rounded-button exercise-button" href="https://leetcode.cn/problems/kth-smallest-element-in-a-bst/solutions/2361685/230-er-cha-sou-suo-shu-zhong-di-k-xiao-d-n3he/" rel="noopener noreferrer" target="_blank">题目解析</a></p>
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