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统计学
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抽样
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推断
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概率论
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计数
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</a>
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概率概念
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分布
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</span>
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</a>
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贝叶斯
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</span>
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</a>
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信息论
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</span>
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机器学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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机器学习
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经典机器学习
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梯度机器学习
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</span>
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</a>
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深度学习
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</span>
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</a>
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强化学习
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</span>
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</a>
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分布式深度学习
|
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</span>
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|
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|
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|
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</a>
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计算语言学
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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语言学基础
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</a>
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嵌入与序列模型
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</span>
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</a>
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Transformer 与语言模型
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</span>
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|
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</a>
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<span class="md-ellipsis">
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高级文本生成
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
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计算机视觉
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计算机视觉
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图像基础
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</span>
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</a>
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目标检测与分割
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多模态学习
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多模态学习
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多模态表征
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</span>
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</a>
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统一多模态架构
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</span>
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自主系统
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感知
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视觉-语言-动作模型
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自动驾驶
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</a>
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太空与极端机器人
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</span>
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图神经网络
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|
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|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2012%3A%20graph%20neural%20networks/04.%20graph%20attention%20networks/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
图注意力网络
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2012%3A%20graph%20neural%20networks/05.%203d%20graph%20networks/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
3D 图网络
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
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|
||
|
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|
||
|
||
|
||
<input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_14" >
|
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|
||
|
||
<label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
计算机与操作系统
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_14">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
计算机与操作系统
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2013%3A%20computing%20and%20OS/01.%20discrete%20maths/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
离散数学
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2013%3A%20computing%20and%20OS/02.%20computer%20architecture/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
计算机体系结构
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2013%3A%20computing%20and%20OS/03.%20operating%20systems/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
操作系统
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2013%3A%20computing%20and%20OS/04.%20concurrency%20and%20parallelism/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
并发与并行
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter%2013%3A%20computing%20and%20OS/05.%20programming%20languages/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
编程语言
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
|
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|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" checked>
|
||
|
||
|
||
<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
数据结构与算法
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="true">
|
||
<label class="md-nav__title" for="__nav_15">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
数据结构与算法
|
||
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
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|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../00.%20foundations/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
基础
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../01.%20arrays%20and%20hashing/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
数组与哈希
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../02.%20linked%20lists%2C%20stacks%2C%20and%20queues/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
链表、栈与队列
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
||
<li class="md-nav__item md-nav__item--active">
|
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|
||
<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
|
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|
||
|
||
|
||
|
||
|
||
<label class="md-nav__link md-nav__link--active" for="__toc">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
树
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<a href="./" class="md-nav__link md-nav__link--active">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
树
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
|
||
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__title" for="__toc">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
目录
|
||
</label>
|
||
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_2" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
二叉树遍历
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="二叉树遍历">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_3" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
简单：二叉树的最大深度
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_4" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
简单：翻转二叉树
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_5" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
中等：二叉树的最近公共祖先
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_6" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
困难：二叉树中的最大路径和
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#bst" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
二叉搜索树（BST）
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="二叉搜索树（BST）">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_7" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
中等：验证二叉搜索树
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#k" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
中等：二叉搜索树中第 K 小的元素
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#trie" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
前缀树（Trie）
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="前缀树（Trie）">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#ii" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
困难：单词搜索 II
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_8" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
并查集（不相交集合）
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="并查集（不相交集合）">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_9" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
中等：连通分量数量
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_10" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
中等：冗余连接
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_11" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
线段树和树状数组
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_12" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
常见陷阱总结
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#neetcode" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
课后练习题（NeetCode）
|
||
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="课后练习题（NeetCode）">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_13" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
二叉树模式
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#bst_1" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
BST 模式
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_14" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
前缀树
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_15" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
|
||
并查集
|
||
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../04.%20graphs/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
图
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../05.%20sorting%20and%20search/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
排序与搜索
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
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|
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生产级软件工程
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SIMD 与 GPU 编程
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ARM 与 NEON
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</span>
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</a>
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x86 与 AVX
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</span>
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</a>
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GPU 架构与 CUDA
|
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</span>
|
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|
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</a>
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<span class="md-ellipsis">
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Triton、TPU 与 Pallas
|
||
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</span>
|
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|
||
|
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|
||
</a>
|
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RISC-V 与嵌入式系统
|
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</span>
|
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|
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|
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|
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</a>
|
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|
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Vulkan Compute 与跨平台 GPU
|
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</span>
|
||
|
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||
|
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AI 推理
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量化
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高效架构
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边缘推理
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扩缩与部署
|
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ML 系统设计
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ML 系统设计
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云计算
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|
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</a>
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大规模基础设施
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|
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</a>
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ML 系统设计
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|
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</span>
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|
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|
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|
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</a>
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前沿 AI
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简单：二叉树的最大深度
|
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中等：二叉树的最近公共祖先
|
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困难：二叉树中的最大路径和
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中等：验证二叉搜索树
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困难：单词搜索 II
|
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并查集（不相交集合）
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中等：连通分量数量
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中等：冗余连接
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线段树和树状数组
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常见陷阱总结
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课后练习题（NeetCode）
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二叉树模式
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BST 模式
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前缀树
|
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</span>
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#_15" class="md-nav__link">
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<span class="md-ellipsis">
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并查集
|
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<div class="md-content" data-md-component="content">
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<article class="md-content__inner md-typeset">
|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
<h1 id="_1">树<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>树是层次化数据结构，是文件系统、数据库、编译器和无数面试题背后的基础。本文件涵盖二叉树、二叉搜索树、平衡树、前缀树、线段树、树状数组和并查集，包括遍历模式、递归思维以及逐步增加难度的题目。</em></p>
|
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<ul>
|
||
<li>
|
||
<p><strong>树</strong>是一个连通的无环图（第13章）。最重要的变体是<strong>二叉树</strong>：每个节点最多有两个子节点（左和右）。树无处不在：编译器中的解析树、浏览器中的 DOM 树、机器学习中的决策树以及数据库中的 B 树。</p>
|
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<li>
|
||
<p>解决树问题的关键洞察：<strong>大多数树问题都可以递归解决</strong>。结构是递归的（树是一个根节点加上两棵子树），因此解法也应是递归的。掌握"解决左子树、解决右子树、合并结果"的模式，你就能解决大多数树问题。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_2">二叉树遍历<a class="headerlink" href="#_2" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p>有四种标准的访问每个节点的方式：</p>
|
||
<ul>
|
||
<li><strong>中序遍历</strong>（左、根、右）：对于 BST，这会按排序顺序访问节点。</li>
|
||
<li><strong>前序遍历</strong>（根、左、右）：用于序列化和复制树。</li>
|
||
<li><strong>后序遍历</strong>（左、右、根）：用于删除和计算大小。</li>
|
||
<li><strong>层序遍历</strong>（BFS）：使用队列逐层访问节点。</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="p">:</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">val</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">left</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">right</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">inorder</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="k">return</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="k">return</span> <span class="n">inorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">]</span> <span class="o">+</span> <span class="n">inorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">preorder</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="k">return</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="k">return</span> <span class="p">[</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">]</span> <span class="o">+</span> <span class="n">preorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span> <span class="o">+</span> <span class="n">preorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="k">def</span><span class="w"> </span><span class="nf">postorder</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a> <span class="k">return</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a> <span class="k">return</span> <span class="n">postorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span> <span class="o">+</span> <span class="n">postorder</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">]</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
|
||
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>
|
||
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="k">def</span><span class="w"> </span><span class="nf">level_order</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="k">return</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a> <span class="n">result</span><span class="p">,</span> <span class="n">queue</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">deque</span><span class="p">([</span><span class="n">root</span><span class="p">])</span>
|
||
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a> <span class="k">while</span> <span class="n">queue</span><span class="p">:</span>
|
||
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a> <span class="n">level</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">queue</span><span class="p">)):</span>
|
||
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
|
||
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a> <span class="n">level</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
|
||
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a> <span class="k">if</span> <span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">:</span>
|
||
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
|
||
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a> <span class="k">if</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">:</span>
|
||
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a> <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a> <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">level</span><span class="p">)</span>
|
||
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a> <span class="k">return</span> <span class="n">result</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>陷阱</strong>：上面的递归遍历在每一步都创建新列表（由于 <code>+</code> 拼接），这是 <span class="arithmatex">\(O(n^2)\)</span>。为了效率，传递一个结果列表并原地追加：</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">inorder_efficient</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">result</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a> <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a> <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="k">if</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="n">inorder_efficient</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a> <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="n">inorder_efficient</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="k">return</span> <span class="n">result</span>
|
||
</code></pre></div>
|
||
<h3 id="_3">简单：二叉树的最大深度<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_depth</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="k">return</span> <span class="mi">0</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="k">return</span> <span class="mi">1</span> <span class="o">+</span> <span class="nb">max</span><span class="p">(</span><span class="n">max_depth</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">),</span> <span class="n">max_depth</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">))</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>递归模式</strong>：基本情况（null → 0），递归子节点，合并（1 + max）。同样的模式适用于数十种树问题。</li>
|
||
</ul>
|
||
<h3 id="_4">简单：翻转二叉树<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">invert_tree</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a> <span class="k">return</span> <span class="kc">None</span>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a> <span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">root</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">invert_tree</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">),</span> <span class="n">invert_tree</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a> <span class="k">return</span> <span class="n">root</span>
|
||
</code></pre></div>
|
||
<h3 id="_5">中等：二叉树的最近公共祖先<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li>
|
||
<p><strong>问题</strong>：找到既是 <span class="arithmatex">\(p\)</span> 又是 <span class="arithmatex">\(q\)</span> 的祖先的最低节点。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>模式</strong>：如果 <span class="arithmatex">\(p\)</span> 和 <span class="arithmatex">\(q\)</span> 都在左子树中，则 LCA 在左子树中。如果都在右子树中，则在右子树中。如果它们分开了（一个在左，一个在右），则当前节点就是 LCA。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">lowest_common_ancestor</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
|
||
<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span> <span class="ow">or</span> <span class="n">root</span> <span class="o">==</span> <span class="n">p</span> <span class="ow">or</span> <span class="n">root</span> <span class="o">==</span> <span class="n">q</span><span class="p">:</span>
|
||
<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a> <span class="k">return</span> <span class="n">root</span>
|
||
<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
|
||
<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a> <span class="n">left</span> <span class="o">=</span> <span class="n">lowest_common_ancestor</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
|
||
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a> <span class="n">right</span> <span class="o">=</span> <span class="n">lowest_common_ancestor</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
|
||
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a>
|
||
<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a> <span class="k">if</span> <span class="n">left</span> <span class="ow">and</span> <span class="n">right</span><span class="p">:</span>
|
||
<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a> <span class="k">return</span> <span class="n">root</span> <span class="c1"># p 和 q 在不同子树中</span>
|
||
<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a> <span class="k">return</span> <span class="n">left</span> <span class="k">if</span> <span class="n">left</span> <span class="k">else</span> <span class="n">right</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>陷阱</strong>：这假设 <span class="arithmatex">\(p\)</span> 和 <span class="arithmatex">\(q\)</span> 都在树中。如果它们可能不在，你需要额外的检查。</li>
|
||
</ul>
|
||
<h3 id="_6">困难：二叉树中的最大路径和<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><strong>问题</strong>：找出任意两个节点之间的最大路径和（路径不需要经过根节点）。</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_path_sum</span><span class="p">(</span><span class="n">root</span><span class="p">):</span>
|
||
<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a> <span class="n">best</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="s1">'-inf'</span><span class="p">)]</span>
|
||
<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>
|
||
<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a> <span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
|
||
<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">node</span><span class="p">:</span>
|
||
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a> <span class="k">return</span> <span class="mi">0</span>
|
||
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a> <span class="n">left</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span> <span class="c1"># 忽略负路径</span>
|
||
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a> <span class="n">right</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>
|
||
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a> <span class="c1"># 经过当前节点的路径（可能作为"转弯点"）</span>
|
||
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a> <span class="n">best</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">best</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span> <span class="o">+</span> <span class="n">left</span> <span class="o">+</span> <span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a>
|
||
<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a> <span class="c1"># 返回到父节点的最大增益</span>
|
||
<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a> <span class="k">return</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span> <span class="o">+</span> <span class="nb">max</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a>
|
||
<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a> <span class="n">dfs</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
|
||
<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a> <span class="k">return</span> <span class="n">best</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>关键洞察</strong>：在每个节点，有两个问题：(1) <em>经过</em>这个节点的最佳路径是什么（左 + 节点 + 右）？(2) 这个节点可以贡献给其<em>父节点</em>的最佳路径是什么（节点 + max(左, 右)，因为路径不能在两个层级分叉）？混淆这两者是最常见的错误。</li>
|
||
</ul>
|
||
<h2 id="bst">二叉搜索树（BST）<a class="headerlink" href="#bst" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li><strong>BST</strong> 满足：对于每个节点，左子树中的所有值都较小，右子树中的所有值都较大。这实现了 <span class="arithmatex">\(O(\log n)\)</span> 的搜索、插入和删除（当平衡时）。</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">search_bst</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
|
||
<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a> <span class="k">return</span> <span class="kc">None</span>
|
||
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a> <span class="k">if</span> <span class="n">target</span> <span class="o"><</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
|
||
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a> <span class="k">return</span> <span class="n">search_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
|
||
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a> <span class="k">elif</span> <span class="n">target</span> <span class="o">></span> <span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
|
||
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a> <span class="k">return</span> <span class="n">search_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
|
||
<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a> <span class="k">else</span><span class="p">:</span>
|
||
<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a> <span class="k">return</span> <span class="n">root</span>
|
||
<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a>
|
||
<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert_bst</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">val</span><span class="p">):</span>
|
||
<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a> <span class="k">return</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
|
||
<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a> <span class="k">if</span> <span class="n">val</span> <span class="o"><</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
|
||
<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a> <span class="n">root</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">insert_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
|
||
<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a> <span class="k">else</span><span class="p">:</span>
|
||
<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a> <span class="n">root</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">insert_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
|
||
<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a> <span class="k">return</span> <span class="n">root</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>陷阱</strong>：BST 操作仅在树平衡时才是 <span class="arithmatex">\(O(\log n)\)</span>。由已排序插入构建的 BST 退化为链表：每次操作 <span class="arithmatex">\(O(n)\)</span>。这就是平衡 BST（AVL、红黑树）存在的原因。</li>
|
||
</ul>
|
||
<h3 id="_7">中等：验证二叉搜索树<a class="headerlink" href="#_7" title="Permanent link">¶</a></h3>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">is_valid_bst</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">lo</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="s1">'-inf'</span><span class="p">),</span> <span class="n">hi</span><span class="o">=</span><span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">)):</span>
|
||
<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">root</span><span class="p">:</span>
|
||
<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a> <span class="k">return</span> <span class="kc">True</span>
|
||
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a> <span class="k">if</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o"><=</span> <span class="n">lo</span> <span class="ow">or</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o">>=</span> <span class="n">hi</span><span class="p">:</span>
|
||
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a> <span class="k">return</span> <span class="kc">False</span>
|
||
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a> <span class="k">return</span> <span class="p">(</span><span class="n">is_valid_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">lo</span><span class="p">,</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">)</span> <span class="ow">and</span>
|
||
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a> <span class="n">is_valid_bst</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">,</span> <span class="n">hi</span><span class="p">))</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>陷阱</strong>：只检查 <code>left.val < root.val < right.val</code> 是错误的。约束条件是左子树中<em>所有</em>节点都更小，而不仅仅是直接子节点。<code>lo</code>/<code>hi</code> 边界将这个约束向下传递。</li>
|
||
</ul>
|
||
<h3 id="k">中等：二叉搜索树中第 K 小的元素<a class="headerlink" href="#k" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><strong>模式</strong>：BST 的中序遍历按排序顺序访问节点。访问的第 <span class="arithmatex">\(k\)</span> 个节点就是答案。</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">kth_smallest</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
|
||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a> <span class="n">count</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a> <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span>
|
||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
|
||
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="k">def</span><span class="w"> </span><span class="nf">inorder</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
|
||
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">node</span> <span class="ow">or</span> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="k">return</span>
|
||
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="n">inorder</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
|
||
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a> <span class="n">count</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="k">if</span> <span class="n">count</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
|
||
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span>
|
||
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a> <span class="k">return</span>
|
||
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a> <span class="n">inorder</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
|
||
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a>
|
||
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a> <span class="n">inorder</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
|
||
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a> <span class="k">return</span> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
</code></pre></div>
|
||
<h2 id="trie">前缀树（Trie）<a class="headerlink" href="#trie" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li><strong>前缀树</strong>逐字符地将字符串存储在树中。每条边代表一个字符，从根到标记节点的路径代表存储的字符串。前缀树实现了 <span class="arithmatex">\(O(L)\)</span> 的查找，其中 <span class="arithmatex">\(L\)</span> 是字符串长度，无论存储了多少个字符串。</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">TrieNode</span><span class="p">:</span>
|
||
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a> <span class="bp">self</span><span class="o">.</span><span class="n">children</span> <span class="o">=</span> <span class="p">{}</span>
|
||
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">is_end</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a>
|
||
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="k">class</span><span class="w"> </span><span class="nc">Trie</span><span class="p">:</span>
|
||
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a> <span class="bp">self</span><span class="o">.</span><span class="n">root</span> <span class="o">=</span> <span class="n">TrieNode</span><span class="p">()</span>
|
||
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a>
|
||
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a> <span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">word</span><span class="p">):</span>
|
||
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a> <span class="n">node</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">root</span>
|
||
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a> <span class="k">for</span> <span class="n">char</span> <span class="ow">in</span> <span class="n">word</span><span class="p">:</span>
|
||
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a> <span class="k">if</span> <span class="n">char</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">:</span>
|
||
<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">[</span><span class="n">char</span><span class="p">]</span> <span class="o">=</span> <span class="n">TrieNode</span><span class="p">()</span>
|
||
<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">[</span><span class="n">char</span><span class="p">]</span>
|
||
<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a> <span class="n">node</span><span class="o">.</span><span class="n">is_end</span> <span class="o">=</span> <span class="kc">True</span>
|
||
<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a>
|
||
<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a> <span class="k">def</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">word</span><span class="p">):</span>
|
||
<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a> <span class="n">node</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">root</span>
|
||
<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a> <span class="k">for</span> <span class="n">char</span> <span class="ow">in</span> <span class="n">word</span><span class="p">:</span>
|
||
<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a> <span class="k">if</span> <span class="n">char</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">:</span>
|
||
<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a> <span class="k">return</span> <span class="kc">False</span>
|
||
<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">[</span><span class="n">char</span><span class="p">]</span>
|
||
<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a> <span class="k">return</span> <span class="n">node</span><span class="o">.</span><span class="n">is_end</span>
|
||
<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a>
|
||
<a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a> <span class="k">def</span><span class="w"> </span><span class="nf">starts_with</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prefix</span><span class="p">):</span>
|
||
<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a> <span class="n">node</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">root</span>
|
||
<a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a> <span class="k">for</span> <span class="n">char</span> <span class="ow">in</span> <span class="n">prefix</span><span class="p">:</span>
|
||
<a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a> <span class="k">if</span> <span class="n">char</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">:</span>
|
||
<a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a> <span class="k">return</span> <span class="kc">False</span>
|
||
<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a> <span class="n">node</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">children</span><span class="p">[</span><span class="n">char</span><span class="p">]</span>
|
||
<a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a> <span class="k">return</span> <span class="kc">True</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>何时使用</strong>：自动补全、拼写检查、单词游戏、IP 路由表。每当你需要基于前缀的操作时。</li>
|
||
</ul>
|
||
<h3 id="ii">困难：单词搜索 II<a class="headerlink" href="#ii" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li>
|
||
<p><strong>问题</strong>：给定一个字符板和一个单词列表，找出所有可以通过遍历相邻单元格形成的单词。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>模式</strong>：从单词列表构建一个前缀树，然后从每个单元格使用前缀树进行 DFS，尽早剪枝分支（如果没有单词以当前前缀开头，则停止）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>陷阱</strong>：没有前缀树的话，你需要为每个单词单独进行 DFS：<span class="arithmatex">\(O(w \cdot m \cdot n \cdot 4^L)\)</span>。前缀树跨单词共享前缀计算，大幅减少了工作量。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="_8">并查集（不相交集合）<a class="headerlink" href="#_8" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li><strong>并查集</strong>跟踪一组不相交集合。两个操作：<code>find(x)</code> 返回 <span class="arithmatex">\(x\)</span> 所在集合的代表元，<code>union(x, y)</code> 合并包含 <span class="arithmatex">\(x\)</span> 和 <span class="arithmatex">\(y\)</span> 的集合。</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">UnionFind</span><span class="p">:</span>
|
||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
|
||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
|
||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">count</span> <span class="o">=</span> <span class="n">n</span> <span class="c1"># 连通分量数</span>
|
||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a>
|
||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a> <span class="k">def</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">!=</span> <span class="n">x</span><span class="p">:</span>
|
||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">[</span><span class="n">x</span><span class="p">])</span> <span class="c1"># 路径压缩</span>
|
||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
|
||
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a>
|
||
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a> <span class="k">def</span><span class="w"> </span><span class="nf">union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
|
||
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a> <span class="n">rx</span><span class="p">,</span> <span class="n">ry</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a> <span class="k">if</span> <span class="n">rx</span> <span class="o">==</span> <span class="n">ry</span><span class="p">:</span>
|
||
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a> <span class="k">return</span> <span class="kc">False</span> <span class="c1"># 已经连通</span>
|
||
<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a> <span class="c1"># 按秩合并</span>
|
||
<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">[</span><span class="n">rx</span><span class="p">]</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">[</span><span class="n">ry</span><span class="p">]:</span>
|
||
<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a> <span class="n">rx</span><span class="p">,</span> <span class="n">ry</span> <span class="o">=</span> <span class="n">ry</span><span class="p">,</span> <span class="n">rx</span>
|
||
<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">[</span><span class="n">ry</span><span class="p">]</span> <span class="o">=</span> <span class="n">rx</span>
|
||
<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">[</span><span class="n">rx</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">[</span><span class="n">ry</span><span class="p">]:</span>
|
||
<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">[</span><span class="n">rx</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a> <span class="bp">self</span><span class="o">.</span><span class="n">count</span> <span class="o">-=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a> <span class="k">return</span> <span class="kc">True</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li>
|
||
<p>通过路径压缩和按秩合并，两个操作都是平摊 <span class="arithmatex">\(O(\alpha(n)) \approx O(1)\)</span>（反阿克曼函数，实际上是常数）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>何时使用</strong>：连通分量、无向图中的环检测、Kruskal 最小生成树、分组等价项。</p>
|
||
</li>
|
||
</ul>
|
||
<h3 id="_9">中等：连通分量数量<a class="headerlink" href="#_9" title="Permanent link">¶</a></h3>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">count_components</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">edges</span><span class="p">):</span>
|
||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a> <span class="n">uf</span> <span class="o">=</span> <span class="n">UnionFind</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
|
||
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a> <span class="n">uf</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a> <span class="k">return</span> <span class="n">uf</span><span class="o">.</span><span class="n">count</span>
|
||
</code></pre></div>
|
||
<h3 id="_10">中等：冗余连接<a class="headerlink" href="#_10" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li>
|
||
<p><strong>问题</strong>：找出从图中移除后使图成为树的那条边（即，创建环的那条边）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>模式</strong>：逐一处理边。第一条两个端点已经在同一分量中的边就是创建环的边。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">find_redundant</span><span class="p">(</span><span class="n">edges</span><span class="p">):</span>
|
||
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a> <span class="n">uf</span> <span class="o">=</span> <span class="n">UnionFind</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
|
||
<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a> <span class="k">if</span> <span class="ow">not</span> <span class="n">uf</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
|
||
<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a> <span class="k">return</span> <span class="p">[</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">]</span> <span class="c1"># 已经连通 → 这条边创建了环</span>
|
||
</code></pre></div>
|
||
<h2 id="_11">线段树和树状数组<a class="headerlink" href="#_11" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li>
|
||
<p><strong>线段树</strong>支持区间查询（子数组上的和、最小值、最大值）和单点更新，两者都是 <span class="arithmatex">\(O(\log n)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>树状数组</strong>（二叉索引树）是前缀和查询和单点更新的更简单、更快的替代方案。它使用一种巧妙的位操作技巧：每个位置存储一个部分和，覆盖范围由最低设置位决定。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">FenwickTree</span><span class="p">:</span>
|
||
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
|
||
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a> <span class="bp">self</span><span class="o">.</span><span class="n">n</span> <span class="o">=</span> <span class="n">n</span>
|
||
<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">tree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>
|
||
<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a> <span class="k">def</span><span class="w"> </span><span class="nf">update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">delta</span><span class="p">):</span>
|
||
<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a> <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span> <span class="c1"># 1-indexed</span>
|
||
<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a> <span class="k">while</span> <span class="n">i</span> <span class="o"><=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
|
||
<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a> <span class="bp">self</span><span class="o">.</span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">delta</span>
|
||
<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a> <span class="n">i</span> <span class="o">+=</span> <span class="n">i</span> <span class="o">&</span> <span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">)</span> <span class="c1"># 加上最低设置位</span>
|
||
<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>
|
||
<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a> <span class="k">def</span><span class="w"> </span><span class="nf">prefix_sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
|
||
<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a> <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a> <span class="n">total</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
|
||
<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a> <span class="n">total</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||
<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a> <span class="n">i</span> <span class="o">-=</span> <span class="n">i</span> <span class="o">&</span> <span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">)</span> <span class="c1"># 移除最低设置位</span>
|
||
<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a> <span class="k">return</span> <span class="n">total</span>
|
||
<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>
|
||
<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a> <span class="k">def</span><span class="w"> </span><span class="nf">range_sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
|
||
<a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">prefix_sum</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">prefix_sum</span><span class="p">(</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">l</span> <span class="o">></span> <span class="mi">0</span> <span class="k">else</span> <span class="mi">0</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<ul>
|
||
<li><strong>何时使用</strong>：需要带更新的重复区间查询的问题。当你只需要前缀和时首选树状数组；当你需要任意区间操作（最小值、最大值、GCD）时使用线段树。</li>
|
||
</ul>
|
||
<hr />
|
||
<h2 id="_12">常见陷阱总结<a class="headerlink" href="#_12" title="Permanent link">¶</a></h2>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>陷阱</th>
|
||
<th>示例</th>
|
||
<th>修复</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>BST 只检查直接子节点</td>
|
||
<td><code>left.val < root.val</code> 遗漏了深层违规</td>
|
||
<td>传递 <code>lo</code>/<code>hi</code> 边界</td>
|
||
</tr>
|
||
<tr>
|
||
<td>递归中 <span class="arithmatex">\(O(n^2)\)</span> 列表拼接</td>
|
||
<td><code>inorder(left) + [val] + inorder(right)</code></td>
|
||
<td>追加到共享列表</td>
|
||
</tr>
|
||
<tr>
|
||
<td>忘记基本情况</td>
|
||
<td>空树上的无限递归</td>
|
||
<td><code>if not root: return</code></td>
|
||
</tr>
|
||
<tr>
|
||
<td>混淆经过路径和到父节点的路径</td>
|
||
<td>最大路径和：在两个层级分叉</td>
|
||
<td>向父节点返回单分支，单独跟踪双分支</td>
|
||
</tr>
|
||
<tr>
|
||
<td>树状数组 1-indexed vs 0-indexed</td>
|
||
<td>树数组中的差一错误</td>
|
||
<td>入口处始终 <code>i += 1</code></td>
|
||
</tr>
|
||
<tr>
|
||
<td>并查集没有路径压缩</td>
|
||
<td>最坏情况下每次 <code>find</code> 是 <span class="arithmatex">\(O(n)\)</span></td>
|
||
<td><code>self.parent[x] = self.find(self.parent[x])</code></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<hr />
|
||
<h2 id="neetcode">课后练习题（NeetCode）<a class="headerlink" href="#neetcode" title="Permanent link">¶</a></h2>
|
||
<h3 id="_13">二叉树模式<a class="headerlink" href="#_13" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><a href="https://neetcode.io/problems/invert-a-binary-tree">翻转二叉树</a> — 基础递归</li>
|
||
<li><a href="https://neetcode.io/problems/depth-of-binary-tree">二叉树的最大深度</a> — 递归深度</li>
|
||
<li><a href="https://neetcode.io/problems/same-binary-tree">相同的树</a> — 同步遍历</li>
|
||
<li><a href="https://neetcode.io/problems/subtree-of-a-binary-tree">另一棵树的子树</a> — 嵌套递归</li>
|
||
<li><a href="https://neetcode.io/problems/level-order-traversal-of-binary-tree">二叉树的层序遍历</a> — 带层级跟踪的 BFS</li>
|
||
<li><a href="https://neetcode.io/problems/binary-tree-maximum-path-sum">二叉树中的最大路径和</a> — 带全局最优的 DFS</li>
|
||
<li><a href="https://neetcode.io/problems/serialize-and-deserialize-binary-tree">序列化与反序列化二叉树</a> — 前序遍历 + null 标记</li>
|
||
</ul>
|
||
<h3 id="bst_1">BST 模式<a class="headerlink" href="#bst_1" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><a href="https://neetcode.io/problems/valid-binary-search-tree">验证二叉搜索树</a> — 边界传播</li>
|
||
<li><a href="https://neetcode.io/problems/kth-smallest-integer-in-bst">二叉搜索树中第 K 小的元素</a> — 中序遍历</li>
|
||
<li><a href="https://neetcode.io/problems/lowest-common-ancestor-in-binary-search-tree">二叉搜索树的最近公共祖先</a> — 利用 BST 排序性质</li>
|
||
</ul>
|
||
<h3 id="_14">前缀树<a class="headerlink" href="#_14" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><a href="https://neetcode.io/problems/implement-prefix-tree">实现 Trie</a> — 基础前缀树操作</li>
|
||
<li><a href="https://neetcode.io/problems/design-word-search-data-structure">设计添加和搜索单词</a> — 前缀树 + 带通配符的 DFS</li>
|
||
<li><a href="https://neetcode.io/problems/search-for-word-ii">单词搜索 II</a> — 前缀树引导的回溯</li>
|
||
</ul>
|
||
<h3 id="_15">并查集<a class="headerlink" href="#_15" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li><a href="https://neetcode.io/problems/count-connected-components">连通分量数量</a> — 基础并查集</li>
|
||
<li><a href="https://neetcode.io/problems/redundant-connection">冗余连接</a> — 通过并查集检测环</li>
|
||
</ul>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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