Unique Binary Search Tree

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

Analyse: Dynamic Planning.

If the root is k, then elements in the left subtree are 1, 2, ... k-1; elements in the right subtree are k+1, k+2, ...n. For the left subtree, root->left can be randomly choose from 1, 2, ...k-1,  so there are k-1 possibilities. So as the right subtree of the root. If f(k) stands for the amount of solution when k is the root node, then f(k) = f(k-1) * f(n-k). 

If there is no node, f(0) = 1; If there is only one node, f(1) = 1.

Runtime: 0ms.

 1 class Solution {
 2 public:
 3     int numTrees(int n) {
 4         vector<int> f(n + 1, 0);
 5         
 6         f[0] = 1;
 7         f[1] = 1;
 8         for(int i = 2; i <=n; i++){
 9             for(int k = 1; k <= i; k++)
10                 f[i] += f[k - 1] * f[i - k];
11         }
12         return f[n];
13     }
14 };