How do you validate that a binary tree is a BST?

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

def is_valid_bst(root):
    def validate(node, low, high):
        if not node:
            return True
        if not (low < node.val < high):
            return False
        return (validate(node.left, low, node.val) and
                validate(node.right, node.val, high))
    return validate(root, float('-inf'), float('inf'))


# --- demo ---
valid = TreeNode(5, TreeNode(3, TreeNode(1), TreeNode(4)), TreeNode(8, None, TreeNode(9)))
invalid = TreeNode(8, TreeNode(4), TreeNode(12, TreeNode(7), None))   # 7 < 8 deep on the right
print(is_valid_bst(valid))     # True
print(is_valid_bst(invalid))   # False