Given n, how many structurally unique BST's (binary search trees) that store values 1 ... n?
Example:
Input: 3 Output: 5 Explanation: 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
LeetCode:链接
首先注意这里是BST而不是普通的Binary Tree,所以数字会对插入的位置有影响。这类找combination/permutation的题都需要找找规律。
n = 0
n = 1
1
n = 2
1 2
\ /
2 1
n = 3
1 3 3 2 1
\ / / / \ \
3 2 1 1 3 2
/ / \ \
2 1 2 3
定义f(n)为unique BST的数量,以n = 3为例:
构造的BST的根节点可以取{1, 2, 3}中的任一数字。
如以1为节点,则left subtree只能有0个节点,而right subtree有2, 3两个节点。所以left/right subtree一共的combination数量为:f(0) * f(2) = 2
以2为节点,则left subtree只能为1,right subtree只能为2:f(1) * f(1) = 1
以3为节点,则left subtree有1, 2两个节点,right subtree有0个节点:f(2)*f(0) = 2
总结规律:
f(0) = 1
f(n) = f(0)*f(n-1) + f(1)*f(n-2) + ... + f(n-2)*f(1) + f(n-1)*f(0)
class Solution(object):
def numTrees(self, n):
"""
:type n: int
:rtype: int
"""
dp = [1, 1, 2]
if n <= 2:
return dp[n]
else:
dp += [0 for i in range(n-2)]
for i in range(3, n+1):
for j in range(1, i+1):
dp[i] += dp[j-1] * dp[i-j]
return dp[n]