Unique Binary Search Trees问题及解法

问题描述：

Given n, how many structurally unique BST's (binary search trees) that store values 1...n?

示例：
Given n = 3, there are a total of 5 unique BST's.

   1         3     3      2      1    \       /     /      / \      \     3     2     1      1   3      2    /     /       \                 \   2     1         2                 3

问题分析：

根据题意分析可得，该题是一类动态规划问题，dp为状态转移数组， 状态转移公式为dp[n] += dp[i - 1]  *  dp[i - k] , 1 <= i <=n, 1 <=k <= i.

过程详见代码：

class Solution {public:int numTrees(int n) {vector<int> dp(n + 1, 0);dp[0] = 1;dp[1] = 1;dp[2] = 2;for (int j = 3; j <= n; j++){for (int i = 0; i < j / 2; i++){dp[j] += 2 * dp[i] * dp[j - 1 - i];}if (j % 2) dp[j] += dp[j / 2] * dp[j / 2];}return dp[n];}};