[LeetCode] 121: Unique Binary Search Trees

[Problem]

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]
方法一

class Solution {
public:
	int numTrees(int n) {
		// Note: The Solution object is instantiated only once and is reused by each test case.

		// invalid
		if(n <= 0)return 0;

		// initial
		int dp[n+1];
		memset(dp, 0, sizeof(dp));
		dp[0] = 1;
		for(int i = 1; i <= n; ++i){
			for(int j = 1; j <= i; ++j){
				dp[i] += dp[j-1]*dp[i-j];
			}
		}
		return dp[n];
	}
};

方法二：Catalan数 C_n+1= C_n * 2(2n+1)/(n+2)

class Solution {
public:
	int numTrees(int n) {
		// Note: The Solution object is instantiated only once and is reused by each test case.

		// invalid
		if(n <= 0)return 0;

		// initial
		int dp[n+1];
		memset(dp, 0, sizeof(dp));
		dp[1] = 1;
		for(int i = 2; i <= n; ++i){
			dp[i] = 2*(2*(i-1)+1)*dp[i-1]/(i-1+2);
		}
		return dp[n];
	}
};

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