from typing import Optional
from trees import Node

type BinarySearchTree[T] = Optional[Node[T]]


def insert[T](node: BinarySearchTree[T], value: T) -> BinarySearchTree[T]:
    prev = None
    curr = node
    while curr:
        prev = curr
        if curr.value > value:
            curr = curr.left
        else:
            curr = curr.right
    if prev is None:
        return Node(value, None, None)
    elif value < prev.value:
        prev.left = Node(value, None, None)
    else:
        prev.right = Node(value, None, None)
    return node


def exists[T](node: Optional[Node[T]],  value: T) -> bool:
    return node and (exists(node.left, value)
                     if value < node.value
                     else (value == node.value or exists(node.right, value)))


def remove[T](node: BinarySearchTree[T], value: T) -> BinarySearchTree[T]:
    if node is None:
        return node
    if value < node.value:
        node.left = remove(node.left, value)
        return node
    if value > node.value:
        node.right = remove(node.right, value)
        return node
    if node.right is None:
        return node.left
    if node.left is None:
        return node.right
    min_node = node.right
    while min_node.left:
        min_node = min_node.left
    node.value = min_node.value
    node.right = remove(node.right, min_node.value)
    return node