问题描述：

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

基本思路：

分别考虑左右子树对应的方法数，然后对于左右子树方法数相乘(左右子树节点数为0的单独处理)。可以递归求子树的个数，也可以用数组记录已知n个节点的子树的方法数。

代码：

    int numTrees(int n) {  //C++
        if(n == 1)
            return 1;
        if(n == 2)
            return 2;
        
        vector<int> record(n+1,0);
        record[0] = 0;
        record[1] = 1;
        record[2] = 2;
        
        
        for(int i = 3; i <= n; i++)
        {
            int sum = 0;
            int k; //left nodes
            for(k = 0; k < i; k++)
            {
                int temp;
                if(k == 0 )
                    temp = record[i-k-1];
                else if(i-k-1 == 0)
                    temp = record[k];
                else temp = record[k]*record[i-k-1];
                sum += temp;
            }
            record[i] = sum;
        }
        return record[n];
    }

[leetcode]Unique Binary Search Trees