[Leetcode 75] 96 Unique Binary Search Trees

Problem:

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

Analysis:

Simple enumeration is not possible. But the problem can be solved by making use of the recursive property of tree and dynamic programming method.

For n = 3, we can choose 1, 2 or 3 as the root of the tree. 

For root is 1, then the tree contains one 2 nodes subtree and a two node tree's unique structure number is 2.

For root is 2, then the tree contains two 1 node subtree and the number of unique structure for each tree is 1, so the final number is 1 * 1 = 1.

For root is 3, the situation is similar as root 1 except the subtree changing from right subtree to left subtree. the number is 2.

So the total number of n =3 is 5.

Other n values can also be computed the same way as mentioned above. Several things to remember:

1. for two subtree with node n1 and n2, the unique of the whole tree is num[n1] * num[n2]

2. due to the symmetric property, we can only go through n/2 values to get the  value

3. pay attention to odd n values. there's an extra item num[n/2] * num[n/2] to be computed.

This DP method only need to go through n/2 values and thus the time complecity is O(n). Since we keep record of all the values that is less than n,  the space requirement is O(n).

Code:

class Solution {
public:
    int numTrees(int n) {
        // Start typing your C/C++ solution below
        // DO NOT write int main() function
        if (n == 0 || n == 1)
            return n;

        int tab[n+1];
        tab[0] = 0;  tab[1] = 1;
        for (int i=2; i<=n; i++) {
            tab[i] = compute(i, tab);
        }
        
        return tab[n];
    }
    
    int compute(int n, int *a) {
        int r = 2 * a[n-1];
        
        for (int i=1; i<n/2; i++) {
            r += 2 * (a[i] * a[n-1-i]);
        }
        
        if (n%2) {
            r += (a[n/2] * a[n/2]);
        }
        
        return r;
    }
};