text: from __future__ import print_function
  # Time:  O(n)
  # Space: O(1)
  #
  # Given a binary tree, determine if it is a valid binary search tree (BST).
  #
  # Assume a BST is defined as follows:
  #
  # The left subtree of a node contains only nodes with keys less than the node's key.
  # The right subtree of a node contains only nodes with keys greater than the node's key.
  # Both the left and right subtrees must also be binary search trees.
  #
  
  # Definition for a  binary tree node
  class TreeNode:
      def __init__(self, x):
          self.val = x
          self.left = None
          self.right = None
  
  # Morris Traversal Solution
  class Solution:
      # @param root, a tree node
      # @return a list of integers
      def isValidBST(self, root):
          prev, cur = None, root
          while cur:
              if cur.left is None:
                  if prev and prev.val >= cur.val:
                      return False
                  prev = cur
                  cur = cur.right
              else:
                  node = cur.left
                  while node.right and node.right != cur:
                      node = node.right
  
                  if node.right is None:
                      node.right = cur
                      cur = cur.left
                  else:
                      if prev and prev.val >= cur.val:
                          return False
                      node.right = None
                      prev = cur
                      cur = cur.right
  
          return True
  
  
  # Time:  O(n)
  # Space: O(h)
  class Solution2:
      # @param root, a tree node
      # @return a boolean
      def isValidBST(self, root):
          return self.isValidBSTRecu(root, float("-inf"), float("inf"))
  
      def isValidBSTRecu(self, root, low, high):
          if root is None:
              return True
  
          return low < root.val and root.val < high \
              and self.isValidBSTRecu(root.left, low, root.val) \
              and self.isValidBSTRecu(root.right, root.val, high)
  
  
  if __name__ == "__main__":
      root = TreeNode(2)
      root.left = TreeNode(1)
      root.right = TreeNode(3)
      print(Solution().isValidBST(root))
  