# LeetCode – Unique Binary Search Trees (Java)

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

Let count[i] be the number of unique binary search trees for i. The number of trees are determined by the number of subtrees which have different root node. For example,

```i=0, count[0]=1 //empty tree

i=1, count[1]=1 //one tree

i=2, count[2]=count[0]*count[1] // 0 is root
+ count[1]*count[0] // 1 is root

i=3, count[3]=count[0]*count[2] // 1 is root
+ count[1]*count[1] // 2 is root
+ count[2]*count[0] // 3 is root

i=4, count[4]=count[0]*count[3] // 1 is root
+ count[1]*count[2] // 2 is root
+ count[2]*count[1] // 3 is root
+ count[3]*count[0] // 4 is root
..
..
..

i=n, count[n] = sum(count[0..k]*count[k+1...n]) 0 <= k < n-1
```

Use dynamic programming to solve the problem.

Java Solution

```public int numTrees(int n) { int[] count = new int[n + 1];   count[0] = 1; count[1] = 1;   for (int i = 2; i <= n; i++) { for (int j = 0; j <= i - 1; j++) { count[i] = count[i] + count[j] * count[i - j - 1]; } }   return count[n]; }```
Category >> Algorithms >> Interview
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(2n)!/(n+1)!n!