From dd2d8a07183af7e848bd94644545c9f3701d882f Mon Sep 17 00:00:00 2001 From: youngyangyang04 <826123027@qq.com> Date: Thu, 8 Oct 2020 10:33:40 +0800 Subject: [PATCH] Update --- README.md | 1 + pics/112.路径总和.png | Bin 102905 -> 107003 bytes pics/112.路径总和1.png | Bin 0 -> 13490 bytes pics/113.路径总和II1.png | Bin 0 -> 26340 bytes problems/0112.路径总和.md | 229 +++++++++++++++++----- problems/0113.路径总和II.md | 2 +- problems/0131.分割回文串.md | 2 +- problems/0257.二叉树的所有路径.md | 2 + 8 files changed, 187 insertions(+), 49 deletions(-) create mode 100644 pics/112.路径总和1.png create mode 100644 pics/113.路径总和II1.png diff --git a/README.md b/README.md index 5d91b0d2..bc3927d1 100644 --- a/README.md +++ b/README.md @@ -97,6 +97,7 @@ * [还在玩耍的你,该总结啦!(本周小结之二叉树)](https://mp.weixin.qq.com/s/QMBUTYnoaNfsVHlUADEzKg) * [二叉树:以为使用了递归,其实还隐藏着回溯](https://mp.weixin.qq.com/s/ivLkHzWdhjQQD1rQWe6zWA) * [二叉树:做了这么多题目了,我的左叶子之和是多少?](https://mp.weixin.qq.com/s/gBAgmmFielojU5Wx3wqFTA) + * 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+在[二叉树:我的左下角的值是多少?](https://mp.weixin.qq.com/s/MH2gbLvzQ91jHPKqiub0Nw)中,因为要遍历树的所有路径,找出深度最深的叶子节点,所以递归函数不要返回值。 + +而本题我们要找一条符合条件的路径,所以递归函数需要返回值,及时返回,那么返回类型是什么呢? 如图所示: @@ -31,41 +43,55 @@ public: 图中可以看出,遍历的路线,并不要遍历整棵树,所以递归函数需要返回值,可以用bool类型表示。 -那么使用深度优先遍历的方式(本题前中后序都可以,无所谓)来遍历二叉树,如下代码我尽量将每一步清晰的表现出来,C++代码如下: +所以代码如下: ``` -class Solution { -private: - bool traversal(TreeNode* cur, int count) { - if (!cur->left && !cur->right && count == 0) return true; // 遇到叶子节点,并且计数为0 - - if (!cur->left && !cur->right) return false; // 遇到叶子节点而没有找到合适的边,直接返回 - - if (cur->left) { // 左 (空节点不遍历) - // 遇到叶子节点返回true,则直接返回true - if (traversal(cur->left, count - cur->left->val)) return true; - } - if (cur->right) { // 右 (空节点不遍历) - // 遇到叶子节点返回true,则直接返回true - if (traversal(cur->right, count - cur->right->val)) return true; - } - return false; - } - -public: - bool hasPathSum(TreeNode* root, int sum) { - if (root == NULL) return false; - return traversal(root, sum - root->val); - } -}; +bool traversal(TreeNode* cur, int count) // 注意函数的返回类型 ``` -那么其实本题一定是有回溯的,没有回溯,如何后撤重新找另一条路径呢,但是貌似以上代码中,**大家貌似没有感受到回溯,那是因为回溯在代码里隐藏起来了。** +2. 确定终止条件 -隐藏在`traversal(cur->left, count - cur->left->val)`这里, 因为把`count - cur->left->val` 直接作为参数传进去,函数结束,count自然恢复到原先的数值了。 +首先计数器如何统计这一条路径的和呢? -为了把回溯的过程体现出来,将`if (traversal(cur->left, count - cur->left->val)) return true;` 改为如下代码: +不要去累加然后判断是否等于目标和,那么代码比较麻烦,可以用递减,让计数器count初始为目标和,然后每次减去遍历路径节点上的数值。 + +如果最后count == 0,同时到了叶子节点的话,说明找到了目标和。 + +如果遍历到了叶子节点,count不为0,就是没找到。 + +递归终止条件代码如下: + +``` +if (!cur->left && !cur->right && count == 0) return true; // 遇到叶子节点,并且计数为0 +if (!cur->left && !cur->right) return false; // 遇到叶子节点而没有找到合适的边,直接返回 +``` + +3. 确定单层递归的逻辑 + +因为终止条件是判断叶子节点,所以递归的过程中就不要让空节点进入递归了。 + +递归函数是有返回值的,如果递归函数返回true,说明找到了合适的路径,应该立刻返回。 + +代码如下: + +``` +if (cur->left) { // 左 (空节点不遍历) + // 遇到叶子节点返回true,则直接返回true + if (traversal(cur->left, count - cur->left->val)) return true; // 注意这里有回溯的逻辑 +} +if (cur->right) { // 右 (空节点不遍历) + // 遇到叶子节点返回true,则直接返回true + if (traversal(cur->right, count - cur->right->val)) return true; // 注意这里有回溯的逻辑 +} +return false; +``` + +以上代码中是包含着回溯的,没有回溯,如何后撤重新找另一条路径呢。 + +回溯隐藏在`traversal(cur->left, count - cur->left->val)`这里, 因为把`count - cur->left->val` 直接作为参数传进去,函数结束,count的数值没有改变。 + +为了把回溯的过程体现出来,可以改为如下代码: ``` if (cur->left) { // 左 @@ -73,9 +99,16 @@ if (cur->left) { // 左 if (traversal(cur->left, count)) return true; count += cur->left->val; // 回溯,撤销处理结果 } +if (cur->right) { // 右 + count -= cur->right->val; + if (traversal(cur->right, count - cur->right->val)) return true; + count += cur->right->val; +} +return false; ``` -这样大家就能感受到回溯了,整体回溯代码如下: + +整体代码如下: ``` class Solution { @@ -105,9 +138,29 @@ public: }; ``` +以上代码精简之后如下: + +``` +class Solution { +public: + bool hasPathSum(TreeNode* root, int sum) { + if (root == NULL) return false; + if (!root->left && !root->right && sum == root->val) { + return true; + } + return hasPathSum(root->left, sum - root->val) || hasPathSum(root->right, sum - root->val); + } +}; +``` + +**是不是发现精简之后的代码,已经完全看不出分析的过程了,所以我们要把题目分析清楚之后,在追求代码精简。** 这一点我已经强调很多次了! + + +## 迭代 + 如果使用栈模拟递归的话,那么如果做回溯呢? -此时栈里一个元素不仅要记录该节点指针,还要记录从头结点到该节点的路径数值总和。 +**此时栈里一个元素不仅要记录该节点指针,还要记录从头结点到该节点的路径数值总和。** C++就我们用pair结构来存放这个栈里的元素。 @@ -146,3 +199,85 @@ public: } }; ``` + +如果大家完全理解了本地的递归方法之后,就可以顺便把leetcode上113. 路径总和II做了。 + +# 113. 路径总和II + +给定一个二叉树和一个目标和,找到所有从根节点到叶子节点路径总和等于给定目标和的路径。 + +说明: 叶子节点是指没有子节点的节点。 + +示例: +给定如下二叉树,以及目标和 sum = 22, + + + + +## 思路 + +113.路径总和II要遍历整个树,找到所有路径,**所以递归函数不要返回值!** + +如图: + + + + +为了尽可能的把细节体现出来,我写出如下代码(**这份代码并不简洁,但是逻辑非常清晰**) + +``` +class Solution { +private: + vector> result; + vector path; + // 递归函数不需要返回值,因为我们要遍历整个树 + void traversal(TreeNode* cur, int count) { + if (!cur->left && !cur->right && count == 0) { // 遇到了叶子节点切找到了和为sum的路径 + result.push_back(path); + return; + } + + if (!cur->left && !cur->right) return ; // 遇到叶子节点而没有找到合适的边,直接返回 + + if (cur->left) { // 左 (空节点不遍历) + path.push_back(cur->left->val); + count -= cur->left->val; + traversal(cur->left, count); // 递归 + count += cur->left->val; // 回溯 + path.pop_back(); // 回溯 + } + if (cur->right) { // 右 (空节点不遍历) + path.push_back(cur->right->val); + count -= cur->right->val; + traversal(cur->right, count); // 递归 + count += cur->right->val; // 回溯 + path.pop_back(); // 回溯 + } + return ; + } + +public: + vector> pathSum(TreeNode* root, int sum) { + result.clear(); + path.clear(); + if (root == NULL) return result; + path.push_back(root->val); // 把根节点放进路径 + traversal(root, sum - root->val); + return result; + } +}; +``` + +至于113. 路径总和II 的迭代法我并没有写,用迭代方式记录所有路径比较麻烦,也没有必要,如果大家感兴趣的话,可以再深入研究研究。 + +# 总结 + +本篇通过leetcode上112. 路径总和 和 113. 路径总和II 详细的讲解了 递归函数什么时候需要返回值,什么不需要返回值。 + +这两道题目是掌握这一知识点非常好的题目,大家看完本篇文章再去做题,就会感受到搜索整棵树和搜索某一路径的差别。 + +对于112. 路径总和,我依然给出了递归法和迭代法,这种题目其实用迭代法会复杂一些,能掌握递归方式就够了! + +今天是长假最后一天了,内容多一些,也是为了尽快让大家恢复学习状态,哈哈。 + +加个油! diff --git a/problems/0113.路径总和II.md b/problems/0113.路径总和II.md index 430e5d70..b12f27f2 100644 --- a/problems/0113.路径总和II.md +++ b/problems/0113.路径总和II.md @@ -18,7 +18,7 @@ 如图: - + 这道题目其实比[112. 路径总和](https://leetcode-cn.com/problems/path-sum/)简单一些,大家做完了本题,可以在做[112. 路径总和](https://leetcode-cn.com/problems/path-sum/)。 diff --git a/problems/0131.分割回文串.md b/problems/0131.分割回文串.md index 1e1244fb..f46a6149 100644 --- a/problems/0131.分割回文串.md +++ b/problems/0131.分割回文串.md @@ -135,7 +135,7 @@ private: } else { // 如果不是则直接跳过 continue; } - backtracking(s, i + 1, path); // 寻找i+1为起始位置的子串 + backtracking(s, i + 1); // 寻找i+1为起始位置的子串 path.pop_back(); // 回溯过程,弹出本次已经填在的子串 } } diff --git a/problems/0257.二叉树的所有路径.md b/problems/0257.二叉树的所有路径.md index 1e083c78..d63fa41d 100644 --- a/problems/0257.二叉树的所有路径.md +++ b/problems/0257.二叉树的所有路径.md @@ -207,6 +207,8 @@ public: 那么在如上代码中,**貌似没有看到回溯的逻辑,其实不然,回溯就隐藏在`traversal(cur->left, path + "->", result);`中的 `path + "->"`。** 每次函数调用完,path依然是没有加上"->" 的,这就是回溯了。 +**如果这里还不理解的话,可以看这篇[二叉树:以为使用了递归,其实还隐藏着回溯](https://mp.weixin.qq.com/s/ivLkHzWdhjQQD1rQWe6zWA),我这这篇中详细的解释了递归中如何隐藏着回溯。 ** + **综合以上,第二种递归的代码虽然精简但把很多重要的点隐藏在了代码细节里,第一种递归写法虽然代码多一些,但是把每一个逻辑处理都完整的展现了出来了。**