5.3.1 Unique Binary Sear Trees

Notes: Given n, how many structurally unique BST's (binary search trees) that store values 1...n? For example, 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  Solution: dp. */解题思路：

n=0，f(0)=0;

n=1,f(1)=1;

n=2,f(2)=f(0)*f(1)+f(1)*f(0)=2;

n=3,f(3)=f(0)*f(2)+f(1)*f(1)+f(2)*f(0)=5;

 所以

class Solution {public:    int numTrees(int n) {        return numTrees_2(n);    }    int numTrees_1(int n) {        int dp[n+1];        memset(dp, 0, sizeof(dp));        dp[0] = 1;        for (int i = 1; i <= n; ++i)            for (int j = 0; j < i; j++)                dp[i] += dp[j] * dp[i-j-1];        return dp[n];    }    int numTrees_2(int n) {        if (n < 0) return 0;        vector<int> dp(n+1, 0);        dp[0] = 1; dp[1] = 1;        for(int i = 2;i <= n; ++i){            dp[i] = dp[i-1] * (4 * i - 2)/(i + 1);        }        return dp[n];            }};