#### Binary Search Tree

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```r"""
A binary search Tree

Example
8
/ \
3   10
/ \    \
1   6    14
/ \   /
4   7 13

>>> t = BinarySearchTree()
>>> t.insert(8, 3, 6, 1, 10, 14, 13, 4, 7)
>>> print(" ".join(repr(i.value) for i in t.traversal_tree()))
8 3 1 6 4 7 10 14 13
>>> print(" ".join(repr(i.value) for i in t.traversal_tree(postorder)))
1 4 7 6 3 13 14 10 8
>>> t.remove(20)
Traceback (most recent call last):
...
>>> BinarySearchTree().search(6)
Traceback (most recent call last):
...
IndexError: Warning: Tree is empty! please use another.

Other example:

>>> testlist = (8, 3, 6, 1, 10, 14, 13, 4, 7)
>>> t = BinarySearchTree()
>>> for i in testlist:
...     t.insert(i)

Prints all the elements of the list in order traversal
>>> print(t)
{'8': ({'3': (1, {'6': (4, 7)})}, {'10': (None, {'14': (13, None)})})}

Test existence
>>> t.search(6) is not None
True
>>> t.search(-1) is not None
False

>>> t.search(6).is_right
True
>>> t.search(1).is_right
False

>>> t.get_max().value
14
>>> t.get_min().value
1
>>> t.empty()
False
>>> for i in testlist:
...     t.remove(i)
>>> t.empty()
True
"""

from collections.abc import Iterable
from typing import Any

class Node:
def __init__(self, value: int | None = None):
self.value = value
self.parent: Node | None = None  # Added in order to delete a node easier
self.left: Node | None = None
self.right: Node | None = None

def __repr__(self) -> str:
from pprint import pformat

if self.left is None and self.right is None:
return str(self.value)
return pformat({f"{self.value}": (self.left, self.right)}, indent=1)

@property
def is_right(self) -> bool:
return self.parent is not None and self is self.parent.right

class BinarySearchTree:
def __init__(self, root: Node | None = None):
self.root = root

def __str__(self) -> str:
"""
Return a string of all the Nodes using in order traversal
"""
return str(self.root)

def __reassign_nodes(self, node: Node, new_children: Node | None) -> None:
if new_children is not None:  # reset its kids
new_children.parent = node.parent
if node.parent is not None:  # reset its parent
if node.is_right:  # If it is the right child
node.parent.right = new_children
else:
node.parent.left = new_children
else:
self.root = new_children

def empty(self) -> bool:
return self.root is None

def __insert(self, value) -> None:
"""
Insert a new node in Binary Search Tree with value label
"""
new_node = Node(value)  # create a new Node
if self.empty():  # if Tree is empty
self.root = new_node  # set its root
else:  # Tree is not empty
parent_node = self.root  # from root
if parent_node is None:
return
while True:  # While we don't get to a leaf
if value < parent_node.value:  # We go left
if parent_node.left is None:
parent_node.left = new_node  # We insert the new node in a leaf
break
else:
parent_node = parent_node.left
else:
if parent_node.right is None:
parent_node.right = new_node
break
else:
parent_node = parent_node.right
new_node.parent = parent_node

def insert(self, *values) -> None:
for value in values:
self.__insert(value)

def search(self, value) -> Node | None:
if self.empty():
raise IndexError("Warning: Tree is empty! please use another.")
else:
node = self.root
# use lazy evaluation here to avoid NoneType Attribute error
while node is not None and node.value is not value:
node = node.left if value < node.value else node.right
return node

def get_max(self, node: Node | None = None) -> Node | None:
"""
We go deep on the right branch
"""
if node is None:
if self.root is None:
return None
node = self.root

if not self.empty():
while node.right is not None:
node = node.right
return node

def get_min(self, node: Node | None = None) -> Node | None:
"""
We go deep on the left branch
"""
if node is None:
node = self.root
if self.root is None:
return None
if not self.empty():
node = self.root
while node.left is not None:
node = node.left
return node

def remove(self, value: int) -> None:
# Look for the node with that label
node = self.search(value)
if node is None:
raise ValueError(msg)

if node.left is None and node.right is None:  # If it has no children
self.__reassign_nodes(node, None)
elif node.left is None:  # Has only right children
self.__reassign_nodes(node, node.right)
elif node.right is None:  # Has only left children
self.__reassign_nodes(node, node.left)
else:
predecessor = self.get_max(
node.left
)  # Gets the max value of the left branch
self.remove(predecessor.value)  # type: ignore
node.value = (
predecessor.value  # type: ignore
)  # Assigns the value to the node to delete and keep tree structure

def preorder_traverse(self, node: Node | None) -> Iterable:
if node is not None:
yield node  # Preorder Traversal
yield from self.preorder_traverse(node.left)
yield from self.preorder_traverse(node.right)

def traversal_tree(self, traversal_function=None) -> Any:
"""
This function traversal the tree.
You can pass a function to traversal the tree as needed by client code
"""
if traversal_function is None:
return self.preorder_traverse(self.root)
else:
return traversal_function(self.root)

def inorder(self, arr: list, node: Node | None) -> None:
"""Perform an inorder traversal and append values of the nodes to
a list named arr"""
if node:
self.inorder(arr, node.left)
arr.append(node.value)
self.inorder(arr, node.right)

def find_kth_smallest(self, k: int, node: Node) -> int:
"""Return the kth smallest element in a binary search tree"""
arr: list[int] = []
self.inorder(arr, node)  # append all values to list using inorder traversal
return arr[k - 1]

def postorder(curr_node: Node | None) -> list[Node]:
"""
postOrder (left, right, self)
"""
node_list = []
if curr_node is not None:
node_list = postorder(curr_node.left) + postorder(curr_node.right) + [curr_node]
return node_list

if __name__ == "__main__":
import doctest

doctest.testmod(verbose=True)
```