Largest BST subtree

Time: O(N); Space: O(H); medium

Given a binary tree, find the largest subtree which is a Binary Search Tree (BST), where largest means subtree with largest number of nodes in it.

Note:

  • A subtree must include all of its descendants.

Example 1:

Input: root = {TreeNode} [10,5,15,1,8,None,7]

    10
   /  \
  5   15
 / \    \
1   8    7

Output: 3

Explanation:

  • The Largest BST Subtree in this case is the highlighted one.

  • The return value is the subtree’s size, which is 3.

Follow up:

  • Can you figure out ways to solve it with O(n) time complexity?

Hints:

  1. You can recursively use algorithm similar to 98.

  2. Validate Binary Search Tree at each node of the tree, which will result in O(NLogN) time complexity.

[6]:

class TreeNode(object):
    def __init__(self, x):
        self.val = x
        self.left = None
        self.right = None

[7]:

class Solution1(object):
    """
    Time: O(N)
    Space: O(H)
    """
    def largestBSTSubtree(self, root):
        """
        :type root: TreeNode
        :rtype: int
        """
        if root is None:
            return 0

        max_size = [1]
        def largestBSTSubtreeHelper(root):
            if root.left is None and root.right is None:
                return 1, root.val, root.val

            left_size, left_min, left_max = 0, root.val, root.val
            if root.left is not None:
                left_size, left_min, left_max = largestBSTSubtreeHelper(root.left)

            right_size, right_min, right_max = 0, root.val, root.val
            if root.right is not None:
                right_size, right_min, right_max = largestBSTSubtreeHelper(root.right)

            size = 0
            if (root.left is None or left_size > 0) and \
               (root.right is None or right_size > 0) and \
                left_max <= root.val <= right_min:

                size = 1 + left_size + right_size

                max_size[0] = max(max_size[0], size)

            return size, left_min, right_max

        largestBSTSubtreeHelper(root)

        return max_size[0]

[8]:

s = Solution1()
root = TreeNode(10)
root.left = TreeNode(5)
root.right = TreeNode(15)
root.left.left = TreeNode(1)
root.left.right = TreeNode(8)
root.right.right = TreeNode(7)
assert s.largestBSTSubtree(root) == 3