Leetcode: Unique Binary Search Trees

// Blog link: https://code.dennyzhang.com/unique-binary-search-trees
// Basic Ideas: dynamic programming
//       Pitfalls: try to compare the values. This direction will make things very complicated
//
//       How to get f(n) from previous values?
//           1. We will have a root, so sum(left sub-tree nodes) + sum(right sub-tree nodes) = n-1
//           2. If root is i, left sub-tree will have i-1 nodes, right sub-tree will have n-k nodes.
//                  How many different types? f(i-1)*f(n-i)
//           3. Loop k from 1 to n. Then collect the total number
//  n = 2:
//
//       1         2
//        \       /
//         2     1
//
//  n = 3:
//
//       1         3     3      2      1
//        \       /     /      / \      \
//         3     2     1      1   3      2
//        /     /       \                 \
//       2     1         2                 3
//
//  n = 4:
//
//
//
// Complexity:

func numTrees(n int) int {
    if n<=1 {return n}
    dp := make([]int, n+1)
    dp[0] = 1
    for i, _:= range dp {
        if i == 0 { continue }
        for j := 0; j<i; j++ {
            dp[i] += dp[j]*dp[i-1-j]
        }
    }
    return dp[n]
}