[Leetcode] DP-- 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

Solution:

 

count the number of ways. use DP.

　　(1) Define the subproblem

    　　dp[i] indicate the number of unique BST rooted at i

 

　　(2) Find the recursion (state transition function)

   　　　 a. there is 0 node, empty tree

                  dp[0] = 1

               b. there is 1 node, just root node

                   dp[1] = 1

              c. there is 2 nodes.   number 1 is rooted,  dp[0]*dp[1];        2 is rooted dp[1]*dp[0]

                   dp[2]  =  dp[0]*dp[1] +  dp[1]*dp[0]

               d. there is 3 nodes.  number 1 is rooted,  dp[0]*dp[2];        2 is rooted dp[1]*dp[1],   3 is rooted dp[2]*dp[0]

                    dp[3]  =  dp[0]*dp[2]  + dp[1]*dp[1]  + dp[2]*dp[0]

 

          Therefore, the transition function is as follows;

              dp[k]   = summation(dp[i] * dp[k-1-i])      for i = 0 to k-1

                                                                          and k from 2 to n

        (3) Get the base case

           dp[0] = 1

           dp[1] = 1

 

       

 dp = [0] * (n+1)
        dp[0] = 1
        dp[1] = 1
        for k in range(2, n+1):
            for i in range(0, k):
                dp[k] += dp[i] * dp[k-1-i]
               # print ("dp k: ", dp[k])
        
        return dp[n]