Leetcode:Unique Binary Search Trees

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

思路：動態規劃的思想，遞歸。具體分析見代碼。

/**
 * Taking 1~n as root respectively:
 *      1 as root: # of trees = F(0) * F(n-1)  // F(0) == 1
 *      2 as root: # of trees = F(1) * F(n-2) 
 *      3 as root: # of trees = F(2) * F(n-3)
 *      ...
 *      n-1 as root: # of trees = F(n-2) * F(1)
 *      n as root:   # of trees = F(n-1) * F(0)
 *
 * So, the formulation is:
 *      F(n) = F(0) * F(n-1) + F(1) * F(n-2) + F(2) * F(n-3) + ... + F(n-2) * F(1) + F(n-1) * F(0)
 */
class Solution {
public:
    int numTrees(int n) {
    int dp[n+1];
    dp[0] = dp[1] = 1;
    for (int i=2; i<=n; i++) {
        dp[i] = 0;
        for (int j=1; j<=i; j++) {
            dp[i] += dp[j-1] * dp[i-j];
        }
    }
    return dp[n];
}
};