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：DP
思路：
G(n): the number of unique BST for a sequence of length n.
F(i, n), 1 <= i <= n: the number of unique BST, where the number i is the root of BST, and the sequence ranges from 1 to n.
G(0)=1, G(1)=1.
G(n) = F(1, n) + F(2, n) + ... + F(n, n).
F(i, n) = G(i-1) * G(n-i) 1 <= i <= n
So,
G(n) = G(0) * G(n-1) + G(1) * G(n-2) + … + G(n-1) * G(0)
Time Complexity: O(N^2) Space Complexity: O(N)
Solution Code：
class Solution {
public int numTrees(int n) {
int [] G = new int[n+1];
G[0] = G[1] = 1;
for(int i=2; i<=n; ++i) {
for(int j=1; j<=i; ++j) {
G[i] += G[j-1] * G[i-j];
}
}
return G[n];
}
}
