From 01977b272670f1778d5a4adb5f5ef3dee25400f2 Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:18:39 +0800 Subject: [PATCH] =?UTF-8?q?Update=20Leetcode=20=E9=A2=98=E8=A7=A3=20-=20?= =?UTF-8?q?=E6=A0=91.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 补充非递归后序遍历代码,补充 LCA、路径和与前缀树等题目的解题思路及复杂度说明;新增非递归遍历和 LCA 示意配图,更新 Trie 配图并补充参考资料。 --- notes/Leetcode 题解 - 树.md | 93 +++++++++++++++++++++---- notes/pics/lc-tree-inorder-stack.png | Bin 0 -> 159011 bytes notes/pics/lc-tree-lca-return.png | Bin 0 -> 117957 bytes notes/pics/lc-tree-postorder-stack.png | Bin 0 -> 238609 bytes notes/pics/lc-tree-trie.png | Bin 150904 -> 151000 bytes 5 files changed, 81 insertions(+), 12 deletions(-) create mode 100644 notes/pics/lc-tree-inorder-stack.png create mode 100644 notes/pics/lc-tree-lca-return.png create mode 100644 notes/pics/lc-tree-postorder-stack.png diff --git a/notes/Leetcode 题解 - 树.md b/notes/Leetcode 题解 - 树.md index 138a3937..014a4607 100644 --- a/notes/Leetcode 题解 - 树.md +++ b/notes/Leetcode 题解 - 树.md @@ -37,6 +37,7 @@ * [Trie](#trie) * [1. 实现一个 Trie](#1-实现一个-trie) * [2. 实现一个 Trie,用来求前缀和](#2-实现一个-trie用来求前缀和) + * [参考资料](#参考资料) @@ -71,7 +72,7 @@ public int maxDepth(TreeNode root) { 15 7 ``` -平衡树左右子树高度差都小于等于 1 +平衡树指的是每个节点的左右子树高度差都小于等于 1。 ```java private boolean result = true; @@ -108,6 +109,8 @@ Input: Return 3, which is the length of the path [4,2,1,3] or [5,2,1,3]. ``` +以某个节点为最高点的最长路径等于它左右子树的高度之和。最长路径不一定经过根节点,因此在求高度的递归中对每个节点都算一次,取最大值。 + ```java private int max = 0; @@ -178,7 +181,7 @@ public TreeNode mergeTrees(TreeNode t1, TreeNode t2) { ### 6. 判断路径和是否等于一个数 -Leetcode : 112. Path Sum (Easy) +112\. Path Sum (Easy) [Leetcode](https://leetcode.com/problems/path-sum/description/) / [力扣](https://leetcode-cn.com/problems/path-sum/description/) @@ -232,6 +235,10 @@ Return 3. The paths that sum to 8 are: 路径不一定以 root 开头,也不一定以 leaf 结尾,但是必须连续。 +对每个节点各统计一次以它开头的路径,每个节点会被它的每个祖先各访问一次,平衡树中时间复杂度为 O(NlogN),退化成链时为 O(N²)。节点值可达 10^9,sum 一路相减可能溢出 int,因此用 long。 + +用[和为 K 的子数组](Leetcode%20题解%20-%20哈希表.md#5-和为-k-的子数组)的前缀和方法可以做到 O(N):哈希表记录根到当前节点路径上各前缀和的出现次数,回溯时减掉。 + ```java public int pathSum(TreeNode root, int sum) { if (root == null) return 0; @@ -335,7 +342,9 @@ private boolean isSymmetric(TreeNode t1, TreeNode t2) { [Leetcode](https://leetcode.com/problems/minimum-depth-of-binary-tree/description/) / [力扣](https://leetcode-cn.com/problems/minimum-depth-of-binary-tree/description/) -树的根节点到叶子节点的最小路径长度 +树的根节点到叶子节点的最小路径长度。 + +需要注意的是,只有一侧子树为空时不能取 min,因为这个节点不是叶子,路径只能走另一侧。 ```java public int minDepth(TreeNode root) { @@ -392,6 +401,8 @@ private boolean isLeaf(TreeNode node){ Output : 2 ``` +做法同[两节点的最长路径](#3-两节点的最长路径),只是 dfs 返回的是节点值都与 root 相同的单侧最长路径,孩子的值与 root 不同时这一侧记为 0。 + ```java private int path = 0; @@ -426,6 +437,8 @@ private int dfs(TreeNode root){ Maximum amount of money the thief can rob = 3 + 3 + 1 = 7. ``` +相连的两个节点不能同时选:选 root 时加上四个孙子子树的结果(val1),不选时为两个孩子子树的结果之和(val2)。同一子树会被祖父和父节点各算一次,因此用 cache 记录。 + ```java Map cache = new HashMap<>(); @@ -461,6 +474,8 @@ Output: 5 一个节点要么具有 0 个或 2 个子节点,如果有子节点,那么根节点是最小的节点。 +孩子的值大于根时,它就是这棵子树中比根大的最小值;等于根时要递归往下找,-1 表示这一侧没有比根大的值。 + ```java public int findSecondMinimumValue(TreeNode root) { if (root == null) return -1; @@ -477,7 +492,7 @@ public int findSecondMinimumValue(TreeNode root) { ## 层次遍历 -使用 BFS 进行层次遍历。不需要使用两个队列来分别存储当前层的节点和下一层的节点,因为在开始遍历一层的节点时,当前队列中的节点数就是当前层的节点数,只要控制遍历这么多节点数,就能保证这次遍历的都是当前层的节点。 +使用 BFS 进行层次遍历。不需要使用两个队列来分别存储当前层的节点和下一层的节点,因为在开始遍历一层的节点时,当前队列中的节点数就是当前层的节点数,只要控制遍历这么多节点数,就能保证这次遍历的都是当前层的节点。按层输出(102 题)和之字形输出的写法见[剑指 Offer 32.2](32.2%20把二叉树打印成多行.md)、[32.3](32.3%20按之字形顺序打印二叉树.md)。 ### 1. 一棵树每层节点的平均数 @@ -527,6 +542,8 @@ Output: 7 ``` +按层遍历时每层先放右孩子、再放左孩子,最后一个出队的节点就是最底层最左边的节点。 + ```java public int findBottomLeftValue(TreeNode root) { Queue queue = new LinkedList<>(); @@ -559,7 +576,7 @@ public int findBottomLeftValue(TreeNode root) { 前序、中序、后序遍历只是在对节点访问的顺序有一点不同,其它都相同。 -① 前序 +前序遍历: ```java void dfs(TreeNode root) { @@ -570,7 +587,7 @@ void dfs(TreeNode root) { } ``` -② 中序 +中序遍历: ```java void dfs(TreeNode root) { @@ -581,7 +598,7 @@ void dfs(TreeNode root) { } ``` -③ 后序 +后序遍历: ```java void dfs(TreeNode root) { @@ -639,12 +656,43 @@ public List postorderTraversal(TreeNode root) { } ``` +上面的写法只是输出顺序正确,节点并不是按后序访问的。要真正按后序访问,可以像[中序遍历](#3-非递归实现二叉树的中序遍历)那样一路向左压栈,栈顶节点的右子树为空或刚访问完(右孩子等于上一个访问的节点 prev)时才弹出访问,否则先转向右子树。没有 prev 时,下图第 6 步会再次转向 5,陷入死循环: + +

+ +```java +public List postorderTraversal(TreeNode root) { + List ret = new ArrayList<>(); + Stack stack = new Stack<>(); + TreeNode cur = root, prev = null; // prev 为上一个访问的节点 + while (cur != null || !stack.isEmpty()) { + while (cur != null) { + stack.push(cur); + cur = cur.left; + } + TreeNode node = stack.peek(); + if (node.right != null && node.right != prev) { + cur = node.right; // 右子树还没访问,先转向右子树 + } else { + stack.pop(); + ret.add(node.val); + prev = node; + } + } + return ret; +} +``` + ### 3. 非递归实现二叉树的中序遍历 94\. Binary Tree Inorder Traversal (Easy) [Leetcode](https://leetcode.com/problems/binary-tree-inorder-traversal/description/) / [力扣](https://leetcode-cn.com/problems/binary-tree-inorder-traversal/description/) +从 cur 一路向左压栈直到为空,弹出栈顶访问后转向它的右子树。以上面的树为例: + +

+ ```java public List inorderTraversal(TreeNode root) { List ret = new ArrayList<>(); @@ -699,7 +747,9 @@ Output: 1 ``` -题目描述:只保留值在 L \~ R 之间的节点 +题目描述:只保留值在 L \~ R 之间的节点。 + +root 的值大于 R 时,它的右子树也都大于 R,结果就是修剪后的左子树;小于 L 时同理。 ```java public TreeNode trimBST(TreeNode root, int L, int R) { @@ -742,7 +792,7 @@ private void inOrder(TreeNode node, int k) { } ``` -递归解法: +递归解法:每层都要重新统计左子树的大小,最坏时间复杂度为 O(N²);节点中保存子树大小时只需沿一条路径往下走。 ```java public int kthSmallest(TreeNode root, int k) { @@ -778,7 +828,7 @@ Output: The root of a Greater Tree like this: 20 13 ``` -先遍历右子树。 +按右、根、左的顺序遍历就是从大到小访问,sum 累加访问过的值并写回当前节点。 ```java private int sum = 0; @@ -815,6 +865,8 @@ private void traver(TreeNode node) { For example, the lowest common ancestor (LCA) of nodes 2 and 8 is 6. Another example is LCA of nodes 2 and 4 is 2, since a node can be a descendant of itself according to the LCA definition. ``` +p、q 分在 root 两侧或其中一个就是 root 时,root 就是最近公共祖先。 + ```java public TreeNode lowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) { if (root.val > p.val && root.val > q.val) return lowestCommonAncestor(root.left, p, q); @@ -841,6 +893,10 @@ public TreeNode lowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) { For example, the lowest common ancestor (LCA) of nodes 5 and 1 is 3. Another example is LCA of nodes 5 and 4 is 5, since a node can be a descendant of itself according to the LCA definition. ``` +lowestCommonAncestor 在 root 子树中查找:同时有 p、q 时返回最近公共祖先,只有其中一个时返回那个节点,都没有时返回 null。左右子树都找到时 p、q 分在 root 两侧,root 就是答案,否则返回找到的那一侧。查找 5 和 4 时遇到 5 就直接返回,因为 4 在 5 的子树中,5 就是答案。 + +

+ ```java public TreeNode lowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) { if (root == null || root == p || root == q) return root; @@ -856,6 +912,8 @@ public TreeNode lowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) { [Leetcode](https://leetcode.com/problems/convert-sorted-array-to-binary-search-tree/description/) / [力扣](https://leetcode-cn.com/problems/convert-sorted-array-to-binary-search-tree/description/) +取中间的元素作为根,两侧的元素个数最多相差 1,因此树是平衡的。 + ```java public TreeNode sortedArrayToBST(int[] nums) { return toBST(nums, 0, nums.length - 1); @@ -889,6 +947,8 @@ One possible answer is: [0,-3,9,-10,null,5], which represents the following heig -10 5 ``` +做法同上一题,只是链表要用快慢指针找中点,并在它前面断开。每层找中点共 O(N),共 O(logN) 层,时间复杂度为 O(NlogN)。 + ```java public TreeNode sortedListToBST(ListNode head) { if (head == null) return null; @@ -936,7 +996,7 @@ Output: True 使用中序遍历得到有序数组之后,再利用双指针对数组进行查找。 -应该注意到,这一题不能用分别在左右子树两部分来处理这种思想,因为两个待求的节点可能分别在左右子树中。 +需要注意的是,这一题不能分别在左右子树中递归求解,因为两个待求的节点可能分别在左右子树中。 ```java public boolean findTarget(TreeNode root, int k) { @@ -1018,6 +1078,8 @@ return [2]. 答案可能不止一个,也就是有多个值出现的次数一样多。 +中序遍历时相同的值连续出现,和前一个节点比较就能算出当前值的次数 curCnt;curCnt 超过 maxCnt 时结果只留当前值,等于时追加。例如依次访问 1、2、2,结果从 [1] 变为 [1, 2],再变为 [2]。 + ```java private int curCnt = 1; private int maxCnt = 1; @@ -1070,7 +1132,7 @@ class Trie { private class Node { Node[] childs = new Node[26]; - boolean isLeaf; + boolean isLeaf; // 是否有单词在这个节点结束,不一定是叶子 } private Node root = new Node(); @@ -1123,6 +1185,8 @@ class Trie { } ``` +需要注意的是,substring 每次都复制剩下的字符串,长度为 L 的单词共复制约 L2 / 2 个字符;改为传入下标或用循环往下走,每个操作是 O(L)。 + ### 2. 实现一个 Trie,用来求前缀和 677\. Map Sum Pairs (Medium) @@ -1189,3 +1253,8 @@ class MapSum { } } ``` + +## 参考资料 + +- [LeetCode 题库](https://leetcode.com/problemset/) +- Sedgewick R, Wayne K. 算法(第 4 版)[M]. 人民邮电出版社, 2012. diff --git a/notes/pics/lc-tree-inorder-stack.png b/notes/pics/lc-tree-inorder-stack.png new file mode 100644 index 0000000000000000000000000000000000000000..e094943a7a3f6e732cbbf9625922849407a1f65f GIT binary patch literal 159011 zcmX_obzGHe)Aa@c>6R7{0a0m?l2l0v5kXM8OIo_5rAxXbr6d)Q?(SAnM7leDXP@W$ zef;53l)d+TT{COeS~GJ#zEG6L!=}PUAP{)6GS6Nj5a{q*)M`vr_)m6qCm;NW$U#!w z;kBNFk=;8zI}5~9m#5sETms>rG;vK()%+qqCMHE z$9jn-K}47@r~Imv{y{Jc=14Y6rUNP6GZaa}92wPMANrAOnRsoIH>laxiN|#Uf3E|l z`;Ap(1?S1#`6?=ETQ@gTFKSm(<`;PtZurn+W^yh#2&t+*O78ceUm(+nW0EabXB8eC zJ-dUM>6~en)Vio^ki{rl&T3DSXO`4$NGv&A+L+109{-#WtzsG%QBYt<(`ieSD(l07 zKjiljUNBNGA!j%{j2@G?DL_mxoJ2BeNkAiR)>lM$5XUEjC789BJ-&C%;fE%>P#Bs7 zY63l`4~w{pQ11JqU#hChE@`}+5`^tHC6!;Pjz{=WG^&zHHYT$KXR??lG53~Vs6W6& z?X-<%ydUi2HS&=uNZ&^$&l}zqp}2Fn=5l_PQcwUdPdNQT^;7@x>LwAIvR|i^Y&2(6 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