Leetcode 96: 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

 

Note : two solutions 1. dfs + memorization  2. dp. We should learn that problems that can be solved in a top - down direction (DFS) we can also usually solve in bottom-up way (DP).

    public int NumTrees(int n) {
        var dp = new int[n + 1];
        dp[0] = 1;
        dp[1] = 1;
        
        for (int nodes = 2; nodes <= n; nodes++)
        {
            for (int i = 1; i <= nodes; i++)
            {
                dp[nodes] += dp[i - 1] * dp[nodes - i];
            }
        }
        
        return dp[n];
    }

 

public class Solution {
    private Dictionary<int, int> cache = new Dictionary<int, int>();
    public int NumTrees(int n) {
        return DFS(1, n);
    }
    
    private int DFS(int start, int end)
    {
        if (start >= end) return 1;
        if (start + 1 == end) return 2;
        
        if (cache.ContainsKey(end - start)) return cache[end - start];
        
        var result = 0;
        for (int i = start; i <= end; i++)
        {
            result += DFS(start, i - 1) * DFS(i + 1, end);    
        }
        
        cache[end - start] = result;
        
        return result;
    }
}