leetcode-96-Unique Binary Search Trees

Given n, how many structurally unique BST’s (binary search trees) that store values 1 … n?

Example:

Input: 3
Output: 5
Explanation:
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

we set dp[n] is the number of BST that store n consecutive values
we can get dp[n] as follows:

we set 1 as root node, so left subtree has 0 children, right subtree has n - 1 children, the number of bst is dp[0] * dpn - 1;

we set 2 as root node, so left subtree has 1 children, right subtree has n - 2 children, the number of bst is dp[1] * dp[n - 2];

generally, we set k as root noode, so left subtree has k - 1 children, right subtree has n - k children, the number of bst is dp[k - 1] * dp[n - k];

dp[n] = d[0] dp[n - 1] + … + dp[k - 1] dp[n - k] + … + dp[n - 1] * dp[0];

class Solution {
public:
    int numTrees(int n) {
        vector<int> dp(n + 1, 0);
        dp[0] = 1;
        for (int i = 1; i <= n; ++i) {
            for (int k = 1; k <= i; ++k) {
                dp[i] += dp[k - 1] * dp[i - k];
            }
        }
        return dp[n];
    }
};