Just apply iterative inorder traversal (Leetcode 94), in a oop design.

class BSTIterator:

    def __init__(self, root: TreeNode):

        self.stack = []
        self.root = root

    def next(self) -> int:
        """
        @return the next smallest number
        """
        while(self.stack or self.root):
            if self.root:
                self.stack.append(self.root)
                self.root = self.root.left
            else:
                node = self.stack[-1]
                self.stack.pop()
                self.root = node.right
                return node.val


    def hasNext(self) -> bool:
        """
        @return whether we have a next smallest number
        """
        return self.stack or self.root