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class="blog-slider__wrp swiper-wrapper" style="transition-duration: 0ms;"><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/波兰式/" title="波兰式"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover6.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/波兰式/" alt="波兰式">波兰式</a><div class="blog-slider__text">波兰式中缀转前缀要将一个中缀表达式转换为前缀表达式，可以通过栈的方式实现。中缀表达式是我们日常书写的算式形式，比如 (A + B) * C，而前缀表达式（也叫波兰表达式）是一种操作符位于操作数前面的表示方法，例如 * + A B C。
转换规则：
操作符的优先级：在转换过程中，优先级需要考虑，例如：
* 和 / 的优先级高于 + 和 -。
括号的优先级最高。
栈的使用：
遇到操作数（数字或变量）时，直接添加到输出。
遇到操作符时，按照优先级将栈中较高优先级的操作符弹出，直到栈顶的操作符优先级低于当前操作符或栈为空，然后将当前操作符压入栈中。
遇到左括号 ( 时，压入栈中。
遇到右括号 ) 时，弹出栈中的操作符直到遇到左括号为止，并将操作符输出。
最终结果：在表达式遍历完毕后，栈中可能会有剩余的操作符，逐一弹出并加入结果中。
解决思路：
步骤一：将中缀表达式转换为后缀表达式（逆波兰表达式）。
步骤二：将后缀表达式再转换为前缀表达式。
其实，将中缀表达式直接转换为前缀表达式与转换为后缀表达式的步骤基本相同，只是操作顺序反转。为了简洁起见，下面是直接从中缀表达式转换为前缀表达式的 ...</div><a class="blog-slider__button" href="2025/12/05/波兰式/" alt="波兰式">详情 </a></div></div><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/动态规划/" title="动态规划"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover5.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/动态规划/" alt="动态规划">动态规划</a><div class="blog-slider__text">首先，虽然动态规划的核心思想就是穷举求最值，但是问题可以千变万化，穷举所有可行解其实并不是一件容易的事，需要你熟练掌握递归思维，只有列出正确的「[[状态转移方程]]」，才能正确地穷举。而且，你需要判断算法问题是否具备「[[最优子结构]]」，是否能够通过子问题的最值得到原问题的最值。另外，动态规划问题存在「重叠子问题」，如果暴力穷举的话效率会很低，所以需要你使用「备忘录」或者「DP table」来优化穷举过程，避免不必要的计算。
框架：123456789101112131415# 自顶向下递归的动态规划def dp(状态1, 状态2, ...): for 选择 in 所有可能的选择: # 此时的状态已经因为做了选择而改变 result = 求最值(result, dp(状态1, 状态2, ...)) return result# 自底向上迭代的动态规划# 初始化 base casedp[0][0][...] = base case# 进行状态转移for 状态1 in 状态1的所有取值： for 状态2 in 状态2的所有取值： ...</div><a class="blog-slider__button" href="2025/12/05/动态规划/" alt="动态规划">详情 </a></div></div><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/二叉树算法纲领/" title="二叉树算法纲领"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover4.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/二叉树算法纲领/" alt="二叉树算法纲领">二叉树算法纲领</a><div class="blog-slider__text">二叉树算法纲领先在开头总结一下，二叉树解题的思维模式分两类：
1、是否可以通过遍历一遍二叉树得到答案？如果可以，用一个 traverse 函数配合外部变量来实现，这叫「遍历」的思维模式。
2、是否可以定义一个递归函数，通过子问题（子树）的答案推导出原问题的答案？如果可以，写出这个递归函数的定义，并充分利用这个函数的返回值，这叫「分解问题」的思维模式。
无论使用哪种思维模式，你都需要思考：
如果单独抽出一个二叉树节点，它需要做什么事情？需要在什么时候（前&#x2F;中&#x2F;后序位置）做？其他的节点不用你操心，递归函数会帮你在所有节点上执行相同的操作。
深入理解前中后序我先甩给你几个问题，请默默思考 30 秒：
1、你理解的二叉树的前中后序遍历是什么，仅仅是三个顺序不同的 List 吗？
2、请分析，后序遍历有什么特殊之处？
3、请分析，为什么多叉树没有中序遍历？
答不上来，说明你对前中后序的理解仅仅局限于教科书，不过没关系，我用类比的方式解释一下我眼中的前中后序遍历。
首先，回顾一下二叉树递归遍历框架：
12345678def traverse(root): if roo ...</div><a class="blog-slider__button" href="2025/12/05/二叉树算法纲领/" alt="二叉树算法纲领">详情 </a></div></div><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/回溯算法/" title="回溯算法"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover3.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/回溯算法/" alt="回溯算法">回溯算法</a><div class="blog-slider__text">二叉树算法纲领先在开头总结一下，二叉树解题的思维模式分两类：
1、是否可以通过遍历一遍二叉树得到答案？如果可以，用一个 traverse 函数配合外部变量来实现，这叫「遍历」的思维模式。
2、是否可以定义一个递归函数，通过子问题（子树）的答案推导出原问题的答案？如果可以，写出这个递归函数的定义，并充分利用这个函数的返回值，这叫「分解问题」的思维模式。
无论使用哪种思维模式，你都需要思考：
如果单独抽出一个二叉树节点，它需要做什么事情？需要在什么时候（前&#x2F;中&#x2F;后序位置）做？其他的节点不用你操心，递归函数会帮你在所有节点上执行相同的操作。
深入理解前中后序我先甩给你几个问题，请默默思考 30 秒：
1、你理解的二叉树的前中后序遍历是什么，仅仅是三个顺序不同的 List 吗？
2、请分析，后序遍历有什么特殊之处？
3、请分析，为什么多叉树没有中序遍历？
答不上来，说明你对前中后序的理解仅仅局限于教科书，不过没关系，我用类比的方式解释一下我眼中的前中后序遍历。
首先，回顾一下二叉树递归遍历框架：
12345678def traverse(root): if roo ...</div><a class="blog-slider__button" href="2025/12/05/回溯算法/" alt="回溯算法">详情 </a></div></div><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/Linux系统配置安装MySQL/" title="Linux系统配置安装MySQL"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover1.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/Linux系统配置安装MySQL/" alt="Linux系统配置安装MySQL">Linux系统配置安装MySQL</a><div class="blog-slider__text">MySQL 安装安装 MySQL对于基于 Debian 的系统（如 Ubuntu）:
12sudo apt updatesudo apt install mysql-server
对于基于 Red Hat 的系统（如 CentOS）:
1sudo yum install mysql-server
安全配置1sudo mysql_secure_installation
这里会让你选密码等级，我选的 2（最高等级）。
细节配置然后会有一系列介绍，全选 y 也没问题，下面是选项的一些解释。
匿名用户默认情况下，MySQL 安装完成后会创建一个匿名用户，这个用户可以无需账户就能登录 MySQL。这通常用于测试或简化安装过程，但如果你打算将 MySQL 用于生产环境，则强烈建议你删除这个匿名用户，以确保安全性。
禁止 root 用户的远程登录MySQL 的 root 用户被设置为只能从 localhost（本地）登录，这是一种安全措施，防止黑客通过网络尝试猜测 root 密码。如果允许 root 远程登录，系统将面临更大的安全风险，尤其是在生产环境中。因此，通常建议禁止 root 的远 ...</div><a class="blog-slider__button" href="2025/12/05/Linux系统配置安装MySQL/" alt="Linux系统配置安装MySQL">详情 </a></div></div><div class="blog-slider__item swiper-slide" style="width: 750px; opacity: 1; transform: translate3d(0px, 0px, 0px); transition-duration: 0ms;"><a class="blog-slider__img" href="2025/12/05/图论/" title="图论"><img width="48" height="48" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover2.png" alt="图片" onerror="this.src=undefined; this.onerror = null;"></a><div class="blog-slider__content"><span class="blog-slider__code">2025-12-05</span><a class="blog-slider__title" href="2025/12/05/图论/" alt="图论">图论</a><div class="blog-slider__text">有向加权图（邻接表实现）我这里给出一个简单的通用实现，后文图论算法教程和习题中可能会用到。其中有一些可以优化的点我写在注释中了。
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192#include &lt;iostream&gt;#include &lt;vector&gt;#include &lt;stdexcept&gt;using namespace std;// 加权有向图的通用实现（邻接表）class WeightedDigraph &#123;public: // 存储相邻节点及边的权重 struct Edge &#123; int to; int weight; Edge(int to, int weight) &#123; ...</div><a class="blog-slider__button" href="2025/12/05/图论/" alt="图论">详情 </a></div></div></div><div class="blog-slider__pagination swiper-pagination-clickable swiper-pagination-bullets"></div></div><div id="topPostGroup"><div class="top-group-list-item"><div class="post_cover left_radius"><a href="2025/12/05/Linux系统配置安装MySQL/" title="Linux系统配置安装MySQL"><span class="top-group-text">荐</span><img class="post_bg" alt="Linux系统配置安装MySQL" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover1.png" onerror="this.src=undefined; this.onerror = null;"></a></div><div class="top-group-info"><a class="article-title" href="2025/12/05/Linux系统配置安装MySQL/" title="Linux系统配置安装MySQL">Linux系统配置安装MySQL</a></div></div><div class="top-group-list-item"><div class="post_cover left_radius"><a href="2025/12/05/二叉树算法纲领/" title="二叉树算法纲领"><span class="top-group-text">荐</span><img class="post_bg" alt="二叉树算法纲领" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover4.png" onerror="this.src=undefined; this.onerror = null;"></a></div><div class="top-group-info"><a class="article-title" href="2025/12/05/二叉树算法纲领/" title="二叉树算法纲领">二叉树算法纲领</a></div></div><div class="top-group-list-item"><div class="post_cover left_radius"><a href="2025/12/05/动态规划/" title="动态规划"><span class="top-group-text">荐</span><img class="post_bg" alt="动态规划" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover5.png" onerror="this.src=undefined; this.onerror = null;"></a></div><div class="top-group-info"><a class="article-title" href="2025/12/05/动态规划/" title="动态规划">动态规划</a></div></div><div class="top-group-list-item"><div class="post_cover left_radius"><a href="2025/12/05/回溯算法/" title="回溯算法"><span class="top-group-text">荐</span><img class="post_bg" alt="回溯算法" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover3.png" onerror="this.src=undefined; this.onerror = null;"></a></div><div class="top-group-info"><a class="article-title" href="2025/12/05/回溯算法/" title="回溯算法">回溯算法</a></div></div><div class="top-group-list-none"></div><div class="top-group-list-none"></div><div class="top-group-list-none"></div></div></div></div></div><script src="https://npm.elemecdn.com/anzhiyu-theme-static@1.0.0/swiper/swiper.min.js"></script><script>function initBlogSlider() {
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</div><div class="category-bar-next" id="category-bar-next" onclick="anzhiyu.scrollCategoryBarToRight()"><i class="anzhiyufont anzhiyu-icon-angle-double-right"></i></div><a class="catalog-more" href="/categories/">更多</a></div></div></div><div class="recent-post-item lastestpost-item" onclick="pjax.loadUrl('/2025/12/05/Linux%E7%B3%BB%E7%BB%9F%E9%85%8D%E7%BD%AE%E5%AE%89%E8%A3%85MySQL/')"><div class="post_cover left"><a href="/2025/12/05/Linux%E7%B3%BB%E7%BB%9F%E9%85%8D%E7%BD%AE%E5%AE%89%E8%A3%85MySQL/" title="Linux系统配置安装MySQL" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover1.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="Linux系统配置安装MySQL" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">数据库系统</div><span class="newPost">最新</span><a class="unvisited-post" href="/2025/12/05/Linux%E7%B3%BB%E7%BB%9F%E9%85%8D%E7%BD%AE%E5%AE%89%E8%A3%85MySQL/" title="Linux系统配置安装MySQL">未读</a></div><a class="article-title" href="/2025/12/05/Linux%E7%B3%BB%E7%BB%9F%E9%85%8D%E7%BD%AE%E5%AE%89%E8%A3%85MySQL/" title="Linux系统配置安装MySQL">Linux系统配置安装MySQL</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:19:15.000Z" title="发表于 2025-12-05 18:19:15" time="2025-12-05 18:19:15">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T11:21:28.668Z" title="更新于 2025-12-05 19:21:28" time="2025-12-05 19:21:28">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E6%95%B0%E6%8D%AE%E5%BA%93%E7%B3%BB%E7%BB%9F/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>数据库系统</span></a></span></div><div class="content">MySQL 安装安装 MySQL对于基于 Debian 的系统（如 Ubuntu）:
12sudo apt updatesudo apt install mysql-server
对于基于 Red Hat 的系统（如 CentOS）:
1sudo yum install mysql-server
安全配置1sudo mysql_secure_installation
这里会让你选密码等级，我选的 2（最高等级）。
细节配置然后会有一系列介绍，全选 y 也没问题，下面是选项的一些解释。
匿名用户默认情况下，MySQL 安装完成后会创建一个匿名用户，这个用户可以无需账户就能登录 MySQL。这通常用于测试或简化安装过程，但如果你打算将 MySQL 用于生产环境，则强烈建议你删除这个匿名用户，以确保安全性。
禁止 root 用户的远程登录MySQL 的 root 用户被设置为只能从 localhost（本地）登录，这是一种安全措施，防止黑客通过网络尝试猜测 root 密码。如果允许 root 远程登录，系统将面临更大的安全风险，尤其是在生产环境中。因此，通常建议禁止 root 的远 ...</div></div></div><div class="recent-post-item" onclick="pjax.loadUrl('/2025/12/05/%E6%B3%A2%E5%85%B0%E5%BC%8F/')"><div class="post_cover left"><a href="/2025/12/05/%E6%B3%A2%E5%85%B0%E5%BC%8F/" title="波兰式" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover6.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="波兰式" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">数据结构</div><a class="unvisited-post" href="/2025/12/05/%E6%B3%A2%E5%85%B0%E5%BC%8F/" title="波兰式">未读</a></div><a class="article-title" href="/2025/12/05/%E6%B3%A2%E5%85%B0%E5%BC%8F/" title="波兰式">波兰式</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:18:54.000Z" title="发表于 2025-12-05 18:18:54" time="2025-12-05 18:18:54">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T10:56:14.363Z" title="更新于 2025-12-05 18:56:14" time="2025-12-05 18:56:14">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>数据结构</span></a></span></div><div class="content">波兰式中缀转前缀要将一个中缀表达式转换为前缀表达式，可以通过栈的方式实现。中缀表达式是我们日常书写的算式形式，比如 (A + B) * C，而前缀表达式（也叫波兰表达式）是一种操作符位于操作数前面的表示方法，例如 * + A B C。
转换规则：
操作符的优先级：在转换过程中，优先级需要考虑，例如：
* 和 / 的优先级高于 + 和 -。
括号的优先级最高。
栈的使用：
遇到操作数（数字或变量）时，直接添加到输出。
遇到操作符时，按照优先级将栈中较高优先级的操作符弹出，直到栈顶的操作符优先级低于当前操作符或栈为空，然后将当前操作符压入栈中。
遇到左括号 ( 时，压入栈中。
遇到右括号 ) 时，弹出栈中的操作符直到遇到左括号为止，并将操作符输出。
最终结果：在表达式遍历完毕后，栈中可能会有剩余的操作符，逐一弹出并加入结果中。
解决思路：
步骤一：将中缀表达式转换为后缀表达式（逆波兰表达式）。
步骤二：将后缀表达式再转换为前缀表达式。
其实，将中缀表达式直接转换为前缀表达式与转换为后缀表达式的步骤基本相同，只是操作顺序反转。为了简洁起见，下面是直接从中缀表达式转换为前缀表达式的 ...</div></div></div><div class="recent-post-item" onclick="pjax.loadUrl('/2025/12/05/%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95/')"><div class="post_cover left"><a href="/2025/12/05/%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95/" title="回溯算法" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover3.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="回溯算法" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">算法</div><a class="unvisited-post" href="/2025/12/05/%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95/" title="回溯算法">未读</a></div><a class="article-title" href="/2025/12/05/%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95/" title="回溯算法">回溯算法</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:18:31.000Z" title="发表于 2025-12-05 18:18:31" time="2025-12-05 18:18:31">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T10:56:38.058Z" title="更新于 2025-12-05 18:56:38" time="2025-12-05 18:56:38">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E7%AE%97%E6%B3%95/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>算法</span></a></span></div><div class="content">二叉树算法纲领先在开头总结一下，二叉树解题的思维模式分两类：
1、是否可以通过遍历一遍二叉树得到答案？如果可以，用一个 traverse 函数配合外部变量来实现，这叫「遍历」的思维模式。
2、是否可以定义一个递归函数，通过子问题（子树）的答案推导出原问题的答案？如果可以，写出这个递归函数的定义，并充分利用这个函数的返回值，这叫「分解问题」的思维模式。
无论使用哪种思维模式，你都需要思考：
如果单独抽出一个二叉树节点，它需要做什么事情？需要在什么时候（前/中/后序位置）做？其他的节点不用你操心，递归函数会帮你在所有节点上执行相同的操作。
深入理解前中后序我先甩给你几个问题，请默默思考 30 秒：
1、你理解的二叉树的前中后序遍历是什么，仅仅是三个顺序不同的 List 吗？
2、请分析，后序遍历有什么特殊之处？
3、请分析，为什么多叉树没有中序遍历？
答不上来，说明你对前中后序的理解仅仅局限于教科书，不过没关系，我用类比的方式解释一下我眼中的前中后序遍历。
首先，回顾一下二叉树递归遍历框架：
12345678def traverse(root): if roo ...</div></div></div><div class="recent-post-item" onclick="pjax.loadUrl('/2025/12/05/%E5%9B%BE%E8%AE%BA/')"><div class="post_cover left"><a href="/2025/12/05/%E5%9B%BE%E8%AE%BA/" title="图论" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover2.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="图论" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">数据结构</div><a class="unvisited-post" href="/2025/12/05/%E5%9B%BE%E8%AE%BA/" title="图论">未读</a></div><a class="article-title" href="/2025/12/05/%E5%9B%BE%E8%AE%BA/" title="图论">图论</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:18:24.000Z" title="发表于 2025-12-05 18:18:24" time="2025-12-05 18:18:24">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T10:56:45.209Z" title="更新于 2025-12-05 18:56:45" time="2025-12-05 18:56:45">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>数据结构</span></a></span></div><div class="content">有向加权图（邻接表实现）我这里给出一个简单的通用实现，后文图论算法教程和习题中可能会用到。其中有一些可以优化的点我写在注释中了。
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192#include <iostream>#include <vector>#include <stdexcept>using namespace std;// 加权有向图的通用实现（邻接表）class WeightedDigraph {public: // 存储相邻节点及边的权重 struct Edge { int to; int weight; Edge(int to, int weight) { ...</div></div></div><div class="recent-post-item" onclick="pjax.loadUrl('/2025/12/05/%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92/')"><div class="post_cover left"><a href="/2025/12/05/%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92/" title="动态规划" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover5.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="动态规划" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">算法</div><a class="unvisited-post" href="/2025/12/05/%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92/" title="动态规划">未读</a></div><a class="article-title" href="/2025/12/05/%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92/" title="动态规划">动态规划</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:17:51.000Z" title="发表于 2025-12-05 18:17:51" time="2025-12-05 18:17:51">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T10:56:25.179Z" title="更新于 2025-12-05 18:56:25" time="2025-12-05 18:56:25">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E7%AE%97%E6%B3%95/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>算法</span></a></span></div><div class="content">首先，虽然动态规划的核心思想就是穷举求最值，但是问题可以千变万化，穷举所有可行解其实并不是一件容易的事，需要你熟练掌握递归思维，只有列出正确的「[[状态转移方程]]」，才能正确地穷举。而且，你需要判断算法问题是否具备「[[最优子结构]]」，是否能够通过子问题的最值得到原问题的最值。另外，动态规划问题存在「重叠子问题」，如果暴力穷举的话效率会很低，所以需要你使用「备忘录」或者「DP table」来优化穷举过程，避免不必要的计算。
框架：123456789101112131415# 自顶向下递归的动态规划def dp(状态1, 状态2, ...): for 选择 in 所有可能的选择: # 此时的状态已经因为做了选择而改变 result = 求最值(result, dp(状态1, 状态2, ...)) return result# 自底向上迭代的动态规划# 初始化 base casedp[0][0][...] = base case# 进行状态转移for 状态1 in 状态1的所有取值： for 状态2 in 状态2的所有取值： ...</div></div></div><div class="recent-post-item" onclick="pjax.loadUrl('/2025/12/05/%E4%BA%8C%E5%8F%89%E6%A0%91%E7%AE%97%E6%B3%95%E7%BA%B2%E9%A2%86/')"><div class="post_cover left"><a href="/2025/12/05/%E4%BA%8C%E5%8F%89%E6%A0%91%E7%AE%97%E6%B3%95%E7%BA%B2%E9%A2%86/" title="二叉树算法纲领" style="display: flex;height: 100%;"><img class="post_bg" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/cover4.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="二叉树算法纲领" style="pointer-events: none"></a></div><div class="recent-post-info"><div class="recent-post-info-top"><div class="recent-post-info-top-tips"><div class="article-categories-original">技能点</div><div class="article-categories-original">算法</div><a class="unvisited-post" href="/2025/12/05/%E4%BA%8C%E5%8F%89%E6%A0%91%E7%AE%97%E6%B3%95%E7%BA%B2%E9%A2%86/" title="二叉树算法纲领">未读</a></div><a class="article-title" href="/2025/12/05/%E4%BA%8C%E5%8F%89%E6%A0%91%E7%AE%97%E6%B3%95%E7%BA%B2%E9%A2%86/" title="二叉树算法纲领">二叉树算法纲领</a></div><div class="article-meta-wrap"><span class="post-meta-date"><i class="anzhiyufont anzhiyu-icon-calendar-days" style="font-size: 15px; display:none"></i><span class="article-meta-label">发表于</span><time datetime="2025-12-05T10:17:09.000Z" title="发表于 2025-12-05 18:17:09" time="2025-12-05 18:17:09">2025-12-05</time><time class="time_hidden" datetime="2025-12-05T10:56:33.052Z" title="更新于 2025-12-05 18:56:33" time="2025-12-05 18:56:33">2025-12-05</time></span><span class="article-meta tags"><a class="article-meta__tags" href="/tags/%E7%AE%97%E6%B3%95/" event.cancelbubble onclick="window.event.cancelBubble=true;"><span> <i class="anzhiyufont anzhiyu-icon-hashtag"></i>算法</span></a></span></div><div class="content">二叉树算法纲领先在开头总结一下，二叉树解题的思维模式分两类：
1、是否可以通过遍历一遍二叉树得到答案？如果可以，用一个 traverse 函数配合外部变量来实现，这叫「遍历」的思维模式。
2、是否可以定义一个递归函数，通过子问题（子树）的答案推导出原问题的答案？如果可以，写出这个递归函数的定义，并充分利用这个函数的返回值，这叫「分解问题」的思维模式。
无论使用哪种思维模式，你都需要思考：
如果单独抽出一个二叉树节点，它需要做什么事情？需要在什么时候（前/中/后序位置）做？其他的节点不用你操心，递归函数会帮你在所有节点上执行相同的操作。
深入理解前中后序我先甩给你几个问题，请默默思考 30 秒：
1、你理解的二叉树的前中后序遍历是什么，仅仅是三个顺序不同的 List 吗？
2、请分析，后序遍历有什么特殊之处？
3、请分析，为什么多叉树没有中序遍历？
答不上来，说明你对前中后序的理解仅仅局限于教科书，不过没关系，我用类比的方式解释一下我眼中的前中后序遍历。
首先，回顾一下二叉树递归遍历框架：
12345678def traverse(root): if roo ...</div></div></div><nav id="pagination"><div class="pagination"><span class="page-number current">1</span><div class="toPageGroup"><input id="toPageText" oninput="value=value.replace(/[^0-9]/g,'')" maxlength="3" onkeyup="this.value=this.value.replace(/[^u4e00-u9fa5w]/g,'')" aria-label="toPage"><a id="toPageButton" onclick="anzhiyu.toPage()"><i class="anzhiyufont anzhiyu-icon-angles-right" style="font-weight: inherit; font-size: 1rem;"></i></a></div></div></nav></div><div class="aside-content" id="aside-content"><div class="card-widget card-info"><div class="card-content"><div class="author-info__sayhi" id="author-info__sayhi" onclick="anzhiyu.changeSayHelloText()"></div><div class="author-info-avatar"><img class="avatar-img" src= "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" onerror="this.onerror=null,this.src="/img/404.jpg"" data-lazy-src="/img/profile.jpg" 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