題目大意

https://leetcode.com/problems/kth-smallest-element-in-a-bst/description/

230. Kth Smallest Element in a BST

Given a binary search tree, write a function kthSmallest to find the kth smallest element in it.

Note:
You may assume k is always valid, 1 ≤ k ≤ BST's total elements.

Example 1:

Input: root = [3,1,4,null,2], k = 13/ \1 4\2 Output: 1

Example 2:

Input: root = [5,3,6,2,4,null,null,1], k = 35/ \3 6/ \2 4/1 Output: 3

Follow up:

What if the BST is modified (insert/delete operations) often and you need to find the kth smallest frequently? How would you optimize the kthSmallest routine?

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給定一棵二叉搜索樹（BST），編寫一個函數kthSmallest找出其中第k小的元素。

注意：
你可以假設k總是有效的， 1 ≤ k ≤ BST的元素總數。

進一步思考：
如果BST的修改（插入/刪除）操作十分頻繁，并且需要頻繁地找出第k小的元素，應該怎樣優化kthSmallest函數？

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解題思路

BST具有如下性質：

• 左子樹中所有元素的值均小于根節點的值
• 右子樹中所有元素的值均大于根節點的值

因此采用中序遍歷（左 -> 根 -> 右）即可以遞增順序訪問BST中的節點，從而得到第k小的元素，時間復雜度O(k)

Python代碼：

# Definition for a binary tree node. class TreeNode(object):def __init__(self, x):self.val = xself.left = Noneself.right = Noneclass Solution(object): def kthSmallest(self, root, k): # 52 ms""":type root: TreeNode:type k: int:rtype: int"""stack = []node = rootwhile node:stack.append(node)node = node.leftx = 1while stack and x <= k:node = stack.pop()x += 1right = node.rightwhile right:stack.append(right)right = right.leftreturn node.val

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遞歸方式：

class Solution(object):def kthSmallest(self, root, k):""":type root: TreeNode:type k: int:rtype: int"""cnt = []self.helper(root, cnt, k)return cnt[k - 1]def helper(self, node, cnt, k):if not node:return Noneself.helper(node.left, cnt, k)cnt.append(node.val)if len(cnt) == k: # 56 ms <= 96msreturn Noneself.helper(node.right, cnt, k)

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進一步思考：
如果BST節點TreeNode的屬性可以擴展，則再添加一個屬性leftCnt，記錄左子樹的節點個數

記當前節點為node 當node不為空時循環： 若k == node.leftCnt + 1：則返回node 否則，若k > node.leftCnt：則令k -= node.leftCnt + 1，令node = node.right 否則，node = node.left

上述算法時間復雜度為O(BST的高度)

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參考

http://bookshadow.com/weblog/2015/07/02/leetcode-kth-smallest-element-bst/

https://leetcode.com/problems/kth-smallest-element-in-a-bst/discuss/63660/3-ways-implemented-in-JAVA-(Python):-Binary-Search-in-order-iterative-and-recursive

轉載于:https://www.cnblogs.com/bymo/p/9667337.html

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