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];        }
};

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