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];        
    }
};