Binary Tree
from __future__ import annotations
class Node:
"""
A Node has data variable and pointers to Nodes to its left and right.
"""
def __init__(self, data: int) -> None:
self.data = data
self.left: Node | None = None
self.right: Node | None = None
def display(tree: Node | None) -> None: # In Order traversal of the tree
"""
>>> root = Node(1)
>>> root.left = Node(0)
>>> root.right = Node(2)
>>> display(root)
0
1
2
>>> display(root.right)
2
"""
if tree:
display(tree.left)
print(tree.data)
display(tree.right)
def depth_of_tree(tree: Node | None) -> int:
"""
Recursive function that returns the depth of a binary tree.
>>> root = Node(0)
>>> depth_of_tree(root)
1
>>> root.left = Node(0)
>>> depth_of_tree(root)
2
>>> root.right = Node(0)
>>> depth_of_tree(root)
2
>>> root.left.right = Node(0)
>>> depth_of_tree(root)
3
>>> depth_of_tree(root.left)
2
"""
return 1 + max(depth_of_tree(tree.left), depth_of_tree(tree.right)) if tree else 0
def is_full_binary_tree(tree: Node) -> bool:
"""
Returns True if this is a full binary tree
>>> root = Node(0)
>>> is_full_binary_tree(root)
True
>>> root.left = Node(0)
>>> is_full_binary_tree(root)
False
>>> root.right = Node(0)
>>> is_full_binary_tree(root)
True
>>> root.left.left = Node(0)
>>> is_full_binary_tree(root)
False
>>> root.right.right = Node(0)
>>> is_full_binary_tree(root)
False
"""
if not tree:
return True
if tree.left and tree.right:
return is_full_binary_tree(tree.left) and is_full_binary_tree(tree.right)
else:
return not tree.left and not tree.right
def main() -> None: # Main function for testing.
tree = Node(1)
tree.left = Node(2)
tree.right = Node(3)
tree.left.left = Node(4)
tree.left.right = Node(5)
tree.left.right.left = Node(6)
tree.right.left = Node(7)
tree.right.left.left = Node(8)
tree.right.left.left.right = Node(9)
print(is_full_binary_tree(tree))
print(depth_of_tree(tree))
print("Tree is: ")
display(tree)
if __name__ == "__main__":
main()# https://en.wikipedia.org/wiki/Tree_traversal
from __future__ import annotations
from dataclasses import dataclass
@dataclass
class Node:
data: int
left: Node | None = None
right: Node | None = None
def make_tree() -> Node:
return Node(1, Node(2, Node(4), Node(5)), Node(3))
def preorder(root: Node):
"""
Pre-order traversal visits root node, left subtree, right subtree.
>>> preorder(make_tree())
[1, 2, 4, 5, 3]
"""
return [root.data] + preorder(root.left) + preorder(root.right) if root else []
def postorder(root: Node):
"""
Post-order traversal visits left subtree, right subtree, root node.
>>> postorder(make_tree())
[4, 5, 2, 3, 1]
"""
return postorder(root.left) + postorder(root.right) + [root.data] if root else []
def inorder(root: Node):
"""
In-order traversal visits left subtree, root node, right subtree.
>>> inorder(make_tree())
[4, 2, 5, 1, 3]
"""
return inorder(root.left) + [root.data] + inorder(root.right) if root else []
def height(root: Node):
"""
Recursive function for calculating the height of the binary tree.
>>> height(None)
0
>>> height(make_tree())
3
"""
return (max(height(root.left), height(root.right)) + 1) if root else 0
def level_order_1(root: Node):
"""
Print whole binary tree in Level Order Traverse.
Level Order traverse: Visit nodes of the tree level-by-level.
"""
if not root:
return
temp = root
que = [temp]
while len(que) > 0:
print(que[0].data, end=" ")
temp = que.pop(0)
if temp.left:
que.append(temp.left)
if temp.right:
que.append(temp.right)
return que
def level_order_2(root: Node, level: int):
"""
Level-wise traversal: Print all nodes present at the given level of the binary tree
"""
if not root:
return root
if level == 1:
print(root.data, end=" ")
elif level > 1:
level_order_2(root.left, level - 1)
level_order_2(root.right, level - 1)
def print_left_to_right(root: Node, level: int):
"""
Print elements on particular level from left to right direction of the binary tree.
"""
if not root:
return
if level == 1:
print(root.data, end=" ")
elif level > 1:
print_left_to_right(root.left, level - 1)
print_left_to_right(root.right, level - 1)
def print_right_to_left(root: Node, level: int):
"""
Print elements on particular level from right to left direction of the binary tree.
"""
if not root:
return
if level == 1:
print(root.data, end=" ")
elif level > 1:
print_right_to_left(root.right, level - 1)
print_right_to_left(root.left, level - 1)
def zigzag(root: Node):
"""
ZigZag traverse: Print node left to right and right to left, alternatively.
"""
flag = 0
height_tree = height(root)
for h in range(1, height_tree + 1):
if flag == 0:
print_left_to_right(root, h)
flag = 1
else:
print_right_to_left(root, h)
flag = 0
def main(): # Main function for testing.
"""
Create binary tree.
"""
root = make_tree()
"""
All Traversals of the binary are as follows:
"""
print(f" In-order Traversal is {inorder(root)}")
print(f" Pre-order Traversal is {preorder(root)}")
print(f"Post-order Traversal is {postorder(root)}")
print(f"Height of Tree is {height(root)}")
print("Complete Level Order Traversal is : ")
level_order_1(root)
print("\nLevel-wise order Traversal is : ")
for h in range(1, height(root) + 1):
level_order_2(root, h)
print("\nZigZag order Traversal is : ")
zigzag(root)
print()
if __name__ == "__main__":
import doctest
doctest.testmod()
main()"""
A binary search Tree
"""
class Node:
def __init__(self, value, parent):
self.value = value
self.parent = parent # Added in order to delete a node easier
self.left = None
self.right = None
def __repr__(self):
from pprint import pformat
if self.left is None and self.right is None:
return str(self.value)
return pformat({"%s" % (self.value): (self.left, self.right)}, indent=1)
class BinarySearchTree:
def __init__(self, root=None):
self.root = root
def __str__(self):
"""
Return a string of all the Nodes using in order traversal
"""
return str(self.root)
def __reassign_nodes(self, node, new_children):
if new_children is not None: # reset its kids
new_children.parent = node.parent
if node.parent is not None: # reset its parent
if self.is_right(node): # If it is the right children
node.parent.right = new_children
else:
node.parent.left = new_children
else:
self.root = new_children
def is_right(self, node):
return node == node.parent.right
def empty(self):
return self.root is None
def __insert(self, value):
"""
Insert a new node in Binary Search Tree with value label
"""
new_node = Node(value, None) # 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
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):
for value in values:
self.__insert(value)
return self
def search(self, value):
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=None):
"""
We go deep on the right branch
"""
if node is None:
node = self.root
if not self.empty():
while node.right is not None:
node = node.right
return node
def get_min(self, node=None):
"""
We go deep on the left branch
"""
if node is None:
node = self.root
if not self.empty():
node = self.root
while node.left is not None:
node = node.left
return node
def remove(self, value):
node = self.search(value) # Look for the node with that label
if node is not None:
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:
tmp_node = self.get_max(
node.left
) # Gets the max value of the left branch
self.remove(tmp_node.value)
node.value = (
tmp_node.value
) # Assigns the value to the node to delete and keep tree structure
def preorder_traverse(self, node):
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):
"""
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):
"""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 = []
self.inorder(arr, node) # append all values to list using inorder traversal
return arr[k - 1]
def postorder(curr_node):
"""
postOrder (left, right, self)
"""
node_list = list()
if curr_node is not None:
node_list = postorder(curr_node.left) + postorder(curr_node.right) + [curr_node]
return node_list
def binary_search_tree():
r"""
Example
8
/ \
3 10
/ \ \
1 6 14
/ \ /
4 7 13
>>> t = BinarySearchTree().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
>>> BinarySearchTree().search(6)
Traceback (most recent call last):
...
IndexError: Warning: Tree is empty! please use another.
"""
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)
if t.search(6) is not None:
print("The value 6 exists")
else:
print("The value 6 doesn't exist")
if t.search(-1) is not None:
print("The value -1 exists")
else:
print("The value -1 doesn't exist")
if not t.empty():
print("Max Value: ", t.get_max().value)
print("Min Value: ", t.get_min().value)
for i in testlist:
t.remove(i)
print(t)
if __name__ == "__main__":
import doctest
doctest.testmod()
# binary_search_tree()"""
This is a python3 implementation of binary search tree using recursion
To run tests:
python -m unittest binary_search_tree_recursive.py
To run an example:
python binary_search_tree_recursive.py
"""
from __future__ import annotations
import unittest
from typing import Iterator
class Node:
def __init__(self, label: int, parent: Node | None) -> None:
self.label = label
self.parent = parent
self.left: Node | None = None
self.right: Node | None = None
class BinarySearchTree:
def __init__(self) -> None:
self.root: Node | None = None
def empty(self) -> None:
"""
Empties the tree
>>> t = BinarySearchTree()
>>> assert t.root is None
>>> t.put(8)
>>> assert t.root is not None
"""
self.root = None
def is_empty(self) -> bool:
"""
Checks if the tree is empty
>>> t = BinarySearchTree()
>>> t.is_empty()
True
>>> t.put(8)
>>> t.is_empty()
False
"""
return self.root is None
def put(self, label: int) -> None:
"""
Put a new node in the tree
>>> t = BinarySearchTree()
>>> t.put(8)
>>> assert t.root.parent is None
>>> assert t.root.label == 8
>>> t.put(10)
>>> assert t.root.right.parent == t.root
>>> assert t.root.right.label == 10
>>> t.put(3)
>>> assert t.root.left.parent == t.root
>>> assert t.root.left.label == 3
"""
self.root = self._put(self.root, label)
def _put(self, node: Node | None, label: int, parent: Node | None = None) -> Node:
if node is None:
node = Node(label, parent)
else:
if label < node.label:
node.left = self._put(node.left, label, node)
elif label > node.label:
node.right = self._put(node.right, label, node)
else:
raise Exception(f"Node with label {label} already exists")
return node
def search(self, label: int) -> Node:
"""
Searches a node in the tree
>>> t = BinarySearchTree()
>>> t.put(8)
>>> t.put(10)
>>> node = t.search(8)
>>> assert node.label == 8
>>> node = t.search(3)
Traceback (most recent call last):
...
Exception: Node with label 3 does not exist
"""
return self._search(self.root, label)
def _search(self, node: Node | None, label: int) -> Node:
if node is None:
raise Exception(f"Node with label {label} does not exist")
else:
if label < node.label:
node = self._search(node.left, label)
elif label > node.label:
node = self._search(node.right, label)
return node
def remove(self, label: int) -> None:
"""
Removes a node in the tree
>>> t = BinarySearchTree()
>>> t.put(8)
>>> t.put(10)
>>> t.remove(8)
>>> assert t.root.label == 10
>>> t.remove(3)
Traceback (most recent call last):
...
Exception: Node with label 3 does not exist
"""
node = self.search(label)
if node.right and node.left:
lowest_node = self._get_lowest_node(node.right)
lowest_node.left = node.left
lowest_node.right = node.right
node.left.parent = lowest_node
if node.right:
node.right.parent = lowest_node
self._reassign_nodes(node, lowest_node)
elif not node.right and node.left:
self._reassign_nodes(node, node.left)
elif node.right and not node.left:
self._reassign_nodes(node, node.right)
else:
self._reassign_nodes(node, None)
def _reassign_nodes(self, node: Node, new_children: Node | None) -> None:
if new_children:
new_children.parent = node.parent
if node.parent:
if node.parent.right == node:
node.parent.right = new_children
else:
node.parent.left = new_children
else:
self.root = new_children
def _get_lowest_node(self, node: Node) -> Node:
if node.left:
lowest_node = self._get_lowest_node(node.left)
else:
lowest_node = node
self._reassign_nodes(node, node.right)
return lowest_node
def exists(self, label: int) -> bool:
"""
Checks if a node exists in the tree
>>> t = BinarySearchTree()
>>> t.put(8)
>>> t.put(10)
>>> t.exists(8)
True
>>> t.exists(3)
False
"""
try:
self.search(label)
return True
except Exception:
return False
def get_max_label(self) -> int:
"""
Gets the max label inserted in the tree
>>> t = BinarySearchTree()
>>> t.get_max_label()
Traceback (most recent call last):
...
Exception: Binary search tree is empty
>>> t.put(8)
>>> t.put(10)
>>> t.get_max_label()
10
"""
if self.root is None:
raise Exception("Binary search tree is empty")
node = self.root
while node.right is not None:
node = node.right
return node.label
def get_min_label(self) -> int:
"""
Gets the min label inserted in the tree
>>> t = BinarySearchTree()
>>> t.get_min_label()
Traceback (most recent call last):
...
Exception: Binary search tree is empty
>>> t.put(8)
>>> t.put(10)
>>> t.get_min_label()
8
"""
if self.root is None:
raise Exception("Binary search tree is empty")
node = self.root
while node.left is not None:
node = node.left
return node.label
def inorder_traversal(self) -> Iterator[Node]:
"""
Return the inorder traversal of the tree
>>> t = BinarySearchTree()
>>> [i.label for i in t.inorder_traversal()]
[]
>>> t.put(8)
>>> t.put(10)
>>> t.put(9)
>>> [i.label for i in t.inorder_traversal()]
[8, 9, 10]
"""
return self._inorder_traversal(self.root)
def _inorder_traversal(self, node: Node | None) -> Iterator[Node]:
if node is not None:
yield from self._inorder_traversal(node.left)
yield node
yield from self._inorder_traversal(node.right)
def preorder_traversal(self) -> Iterator[Node]:
"""
Return the preorder traversal of the tree
>>> t = BinarySearchTree()
>>> [i.label for i in t.preorder_traversal()]
[]
>>> t.put(8)
>>> t.put(10)
>>> t.put(9)
>>> [i.label for i in t.preorder_traversal()]
[8, 10, 9]
"""
return self._preorder_traversal(self.root)
def _preorder_traversal(self, node: Node | None) -> Iterator[Node]:
if node is not None:
yield node
yield from self._preorder_traversal(node.left)
yield from self._preorder_traversal(node.right)
class BinarySearchTreeTest(unittest.TestCase):
@staticmethod
def _get_binary_search_tree() -> BinarySearchTree:
r"""
8
/ \
3 10
/ \ \
1 6 14
/ \ /
4 7 13
\
5
"""
t = BinarySearchTree()
t.put(8)
t.put(3)
t.put(6)
t.put(1)
t.put(10)
t.put(14)
t.put(13)
t.put(4)
t.put(7)
t.put(5)
return t
def test_put(self) -> None:
t = BinarySearchTree()
assert t.is_empty()
t.put(8)
r"""
8
"""
assert t.root is not None
assert t.root.parent is None
assert t.root.label == 8
t.put(10)
r"""
8
\
10
"""
assert t.root.right is not None
assert t.root.right.parent == t.root
assert t.root.right.label == 10
t.put(3)
r"""
8
/ \
3 10
"""
assert t.root.left is not None
assert t.root.left.parent == t.root
assert t.root.left.label == 3
t.put(6)
r"""
8
/ \
3 10
\
6
"""
assert t.root.left.right is not None
assert t.root.left.right.parent == t.root.left
assert t.root.left.right.label == 6
t.put(1)
r"""
8
/ \
3 10
/ \
1 6
"""
assert t.root.left.left is not None
assert t.root.left.left.parent == t.root.left
assert t.root.left.left.label == 1
with self.assertRaises(Exception):
t.put(1)
def test_search(self) -> None:
t = self._get_binary_search_tree()
node = t.search(6)
assert node.label == 6
node = t.search(13)
assert node.label == 13
with self.assertRaises(Exception):
t.search(2)
def test_remove(self) -> None:
t = self._get_binary_search_tree()
t.remove(13)
r"""
8
/ \
3 10
/ \ \
1 6 14
/ \
4 7
\
5
"""
assert t.root is not None
assert t.root.right is not None
assert t.root.right.right is not None
assert t.root.right.right.right is None
assert t.root.right.right.left is None
t.remove(7)
r"""
8
/ \
3 10
/ \ \
1 6 14
/
4
\
5
"""
assert t.root.left is not None
assert t.root.left.right is not None
assert t.root.left.right.left is not None
assert t.root.left.right.right is None
assert t.root.left.right.left.label == 4
t.remove(6)
r"""
8
/ \
3 10
/ \ \
1 4 14
\
5
"""
assert t.root.left.left is not None
assert t.root.left.right.right is not None
assert t.root.left.left.label == 1
assert t.root.left.right.label == 4
assert t.root.left.right.right.label == 5
assert t.root.left.right.left is None
assert t.root.left.left.parent == t.root.left
assert t.root.left.right.parent == t.root.left
t.remove(3)
r"""
8
/ \
4 10
/ \ \
1 5 14
"""
assert t.root is not None
assert t.root.left.label == 4
assert t.root.left.right.label == 5
assert t.root.left.left.label == 1
assert t.root.left.parent == t.root
assert t.root.left.left.parent == t.root.left
assert t.root.left.right.parent == t.root.left
t.remove(4)
r"""
8
/ \
5 10
/ \
1 14
"""
assert t.root.left is not None
assert t.root.left.left is not None
assert t.root.left.label == 5
assert t.root.left.right is None
assert t.root.left.left.label == 1
assert t.root.left.parent == t.root
assert t.root.left.left.parent == t.root.left
def test_remove_2(self) -> None:
t = self._get_binary_search_tree()
t.remove(3)
r"""
8
/ \
4 10
/ \ \
1 6 14
/ \ /
5 7 13
"""
assert t.root is not None
assert t.root.left is not None
assert t.root.left.left is not None
assert t.root.left.right is not None
assert t.root.left.right.left is not None
assert t.root.left.right.right is not None
assert t.root.left.label == 4
assert t.root.left.right.label == 6
assert t.root.left.left.label == 1
assert t.root.left.right.right.label == 7
assert t.root.left.right.left.label == 5
assert t.root.left.parent == t.root
assert t.root.left.right.parent == t.root.left
assert t.root.left.left.parent == t.root.left
assert t.root.left.right.left.parent == t.root.left.right
def test_empty(self) -> None:
t = self._get_binary_search_tree()
t.empty()
assert t.root is None
def test_is_empty(self) -> None:
t = self._get_binary_search_tree()
assert not t.is_empty()
t.empty()
assert t.is_empty()
def test_exists(self) -> None:
t = self._get_binary_search_tree()
assert t.exists(6)
assert not t.exists(-1)
def test_get_max_label(self) -> None:
t = self._get_binary_search_tree()
assert t.get_max_label() == 14
t.empty()
with self.assertRaises(Exception):
t.get_max_label()
def test_get_min_label(self) -> None:
t = self._get_binary_search_tree()
assert t.get_min_label() == 1
t.empty()
with self.assertRaises(Exception):
t.get_min_label()
def test_inorder_traversal(self) -> None:
t = self._get_binary_search_tree()
inorder_traversal_nodes = [i.label for i in t.inorder_traversal()]
assert inorder_traversal_nodes == [1, 3, 4, 5, 6, 7, 8, 10, 13, 14]
def test_preorder_traversal(self) -> None:
t = self._get_binary_search_tree()
preorder_traversal_nodes = [i.label for i in t.preorder_traversal()]
assert preorder_traversal_nodes == [8, 3, 1, 6, 4, 5, 7, 10, 14, 13]
def binary_search_tree_example() -> None:
r"""
Example
8
/ \
3 10
/ \ \
1 6 14
/ \ /
4 7 13
\
5
Example After Deletion
4
/ \
1 7
\
5
"""
t = BinarySearchTree()
t.put(8)
t.put(3)
t.put(6)
t.put(1)
t.put(10)
t.put(14)
t.put(13)
t.put(4)
t.put(7)
t.put(5)
print(
"""
8
/ \\
3 10
/ \\ \\
1 6 14
/ \\ /
4 7 13
\\
5
"""
)
print("Label 6 exists:", t.exists(6))
print("Label 13 exists:", t.exists(13))
print("Label -1 exists:", t.exists(-1))
print("Label 12 exists:", t.exists(12))
# Prints all the elements of the list in inorder traversal
inorder_traversal_nodes = [i.label for i in t.inorder_traversal()]
print("Inorder traversal:", inorder_traversal_nodes)
# Prints all the elements of the list in preorder traversal
preorder_traversal_nodes = [i.label for i in t.preorder_traversal()]
print("Preorder traversal:", preorder_traversal_nodes)
print("Max. label:", t.get_max_label())
print("Min. label:", t.get_min_label())
# Delete elements
print("\nDeleting elements 13, 10, 8, 3, 6, 14")
print(
"""
4
/ \\
1 7
\\
5
"""
)
t.remove(13)
t.remove(10)
t.remove(8)
t.remove(3)
t.remove(6)
t.remove(14)
# Prints all the elements of the list in inorder traversal after delete
inorder_traversal_nodes = [i.label for i in t.inorder_traversal()]
print("Inorder traversal after delete:", inorder_traversal_nodes)
# Prints all the elements of the list in preorder traversal after delete
preorder_traversal_nodes = [i.label for i in t.preorder_traversal()]
print("Preorder traversal after delete:", preorder_traversal_nodes)
print("Max. label:", t.get_max_label())
print("Min. label:", t.get_min_label())
if __name__ == "__main__":
binary_search_tree_example()Binary Trees
Explain and implement a Binary Tree.
A tree is a collection of nodes and edges between them.
It cannot have any cycles, which are edges that form a loop between nodes.
We also only consider rooted trees in computer science, which is a tree that has one root node that is able to access all other nodes.
For a tree to be a binary tree, each node can have a maximum of two children.
It's important to be able to identify and explain tree terminology as well. If given a tree, be able to point out each component.
root: The single node of a tree that can access every other node through edges.
parent node: A node that is connected to lower nodes in the tree. If a tree only has one node, it is not a parent node because there are no children.
child node: A node that is connected to a higher node in the tree. Every node except for the root is a child node of some parent.
sibling nodes: Nodes that have the same parent.
leaf node: A node that has no children (at the ends of the branches of the tree)
internal node: A non-leaf node (aka a parent)
path: A series of nodes that can be traveled through edges.
subtree: A smaller portion of the original tree. Any node that is not the root node is itself the root of a subtree.
Know the basics of each term
A non-empty tree has to have a root.
A tree doesn't have any parent nodes if there are no children.
What's the min/max number of parent and leaf nodes for a tree with 5 nodes?
Implementing as a balanced tree results in min number of parents and max number of leaves: 2 parents, 3 leaves
All that we need in order to implement a binary tree is a TreeNode class that can store a value and references to a left and right child. We can create a tree by assigning the left and right properties to point to other TreeNode instances:
class TreeNode {
constructor(val) {
this.val = val;
this.left = null;
this.right = null;
}
}Identify the three types of tree traversals: pre-order, in-order, and post-order.
Pre-order: Values are accessed as soon as the node is reached.
In-order: Values are accessed after we have fully explored the left but before we explore the right branch.
Post-order: Values are accessed after all of our children have been accessed.
*Breadth First: The previous three are types of Depth First Traversals. Breadth first accesses values of nodes by level, left to right, top to bottom.
Breadth First: 20, 9, 24, 7, 11, 23, 27, 3, 10, 17, 36, 30
Pre-order: 20, 9, 7, 3, 11, 10, 17, 24, 23, 27, 36, 30
In-order: 3, 7, 9, 10, 11, 17, 20, 23, 24, 27, 30, 36
Post-order: 3, 7, 10, 17, 11, 9, 23, 30, 36, 27, 24, 20
Explain and implement a Binary Search Tree.
A binary search tree is a binary tree with the added stipulation that all values to the left of a node are less than its value and all values to the right are greater than its value.
Example of a BST with an insert method. You won't be asked to implement a removal:
class BST {
constructor() {
this.root = null;
}
insert(val, currentNode=this.root) {
if(!this.root) {
this.root = new TreeNode(val);
return;
}
if (val < currentNode.val) {
if (!currentNode.left) {
currentNode.left = new TreeNode(val);
} else {
this.insert(val, currentNode.left);
}
} else {
if (!currentNode.right) {
currentNode.right = new TreeNode(val);
} else {
this.insert(val, currentNode.right);
}
}
}
}
# Implement a Binary Search Tree (BST) that can insert values and check if
# values are present
class Node(object):
def __init__(self, value):
self.value = value
self.left = None
self.right = None
class BST(object):
def __init__(self, root):
self.root = Node(root)
def insert(self, new_val):
if(self.root.left==None):
if(self.root.value>new_val):
self.root.left = Node(new_val)
elif(self.root.right==None):
if(self.root.value<new_val):
self.root.right = Node(new_val)
else:
current = self.root
while(current.left!=None or current.right!=None):
if(current.value>new_val):
current = current.left
else:
current = current.right
if(current.left==None):
current.left = Node(new_val)
else:
current.right = Node(new_val)
def search(self, find_val):
if(self.root.left==None and self.root.right==None and self.root.value!=find_val):
return False
else:
current = self.root
val_possible = True
while(val_possible):
if(current.value==find_val):
return True
if(current.value<find_val):
current = current.right
else:
current = current.left
if(current==None):
return False
if(current.value<find_val and (current.right==None or current.right>find_val)):
return False
if(current.value>find_val and (current.left==None or current.left<find_val)):
return False
# Set up tree
tree = BST(4)
# Insert elements
tree.insert(2)
tree.insert(1)
tree.insert(3)
tree.insert(5)
# Check search
# Should be True
print tree.search(4)
# Should be False
print tree.search(6)
class Solution(object):
def topKFrequent(self, nums, k):
number_frequency = {}
frequency_list = {}
for i in nums:
if i not in number_frequency:
number_frequency[i] = 1
else:
number_frequency[i] += 1
for key, value in number_frequency.items():
if value not in frequency_list:
frequency_list[value] = [key]
else:
frequency_list[value].append(key)
result = []
for i in range(len(nums), 0, -1):
if i in frequency_list:
result.extend(frequency_list[i])
if len(result) >= k:
break
return result
ob1 = Solution()
print(ob1.topKFrequent([1, 1, 1, 1, 2, 2, 3, 3, 3], 2))
Balanced Binary Tree
Balanced Binary Tree
Given a binary tree class that looks like this:
class BinaryTreeNode {
constructor(value) {
this.value = value;
this.left = null;
this.right = null;
}
insertLeft(value) {
this.left = new BinaryTreeNode(value);
return this.left;
}
insertRight(value) {
this.right = new BinaryTreeNode(value);
return this.right;
}
}write a function that checks to see if a given binary tree is perfectly balanced, meaning all leaf nodes are located at the same depth. Your function should return true if the tree is perfectly balanced and false otherwise.
Analyze the time and space complexity of your function.
JS Solution:
/*
A recursive solution
How would you solve this iteratively?
*/
const checkBalanced = (rootNode) => {
// An empty tree is balanced by default
if (!rootNode) return true;
// recursive helper function to check the min depth of the tree
const minDepth = (node) => {
if (!node) return 0;
return 1 + Math.min(minDepth(node.left), minDepth(node.right));
};
// recursive helper function to check the max depth of the tree
const maxDepth = (node) => {
if (!node) return 0;
return 1 + Math.max(maxDepth(node.left), maxDepth(node.right));
};
return maxDepth(rootNode) - minDepth(rootNode) === 0;
};
/* Some console.log tests */
class BinaryTreeNode {
constructor(value) {
this.value = value;
this.left = null;
this.right = null;
}
insertLeft(value) {
this.left = new BinaryTreeNode(value);
return this.left;
}
insertRight(value) {
this.right = new BinaryTreeNode(value);
return this.right;
}
}
const root = new BinaryTreeNode(5);
console.log(checkBalanced(root)); // should print true
root.insertLeft(10);
console.log(checkBalanced(root)); // should print false
root.insertRight(11);
console.log(checkBalanced(root)); // should print true;
# A recursive solution
# How would you solve this iteratively?
def checkBalanced(rootNode):
# An empty tree is balanced by default
if rootNode == None:
return True
# recursive helper function to check the min depth of the tree
def minDepth(node):
if node == None:
return 0
return 1 + min(minDepth(node.left), minDepth(node.right))
# recursive helper function to check the max depth of the tree
def maxDepth(node):
if node == None:
return 0
return 1 + max(maxDepth(node.left), maxDepth(node.right))
return maxDepth(rootNode) - minDepth(rootNode) == 0
# Some console.log tests
class BinaryTreeNode:
def __init__(self, value):
self.value = value
self.left = None
self.right = None
def insertLeft(self, value):
self.left = BinaryTreeNode(value)
return self.left
def insertRight(self, value):
self.right = BinaryTreeNode(value)
return self.right
root = BinaryTreeNode(5)
print(checkBalanced(root)) # should print True
root.insertLeft(10)
print(checkBalanced(root)) # should print False
root.insertRight(11)
print(checkBalanced(root)) # should print True
Binary Search Tree from Sorted Array
Given an array that is sorted in ascending order containing unique integer elements, write a function that receives the sorted array as input and creates a valid binary search tree with minimal height.
For example, given an array [1, 2, 3, 4, 5, 6, 7], your function should return a binary search tree with the form
4
/ \
2 6
/ \ / \
1 3 5 7Note that when we say "binary search tree" in this case, we're just talking about a tree that exhibits the expected form of a binary search tree. The tree in this case won't have an insert method that does the work of receiving a value and then inserting it in a valid spot in the binary search tree. Your function should place the values in valid spots that adhere to the rules of binary search trees, while also seeking to minimize the overall height of the tree.
Here's a BinaryTreeNode class that you can use to construct a binary search tree:
class BinaryTreeNode:
def __init__(self, value):
self.value = value
self.left = None
self.right = NoneAnalyze the time and space complexity of your solution.
Create a Minimal Height BST from Sorted Array
Understanding the Problem
This problem asks us to create a valid binary search tree from a sorted array of integers. More specifically, the resulting binary search tree needs to be of minimal height. Our function should return the root node of the created binary search tree.
From the given example where the input is [1, 2, 3, 4, 5, 6, 7], the expected answer is a binary search tree of height 3. This is the minimal height that can be achieved for an array of 7 seven elements. Try as we might, there's no way to construct a binary search tree containing all of these elements that has a shorter height.
Coming Up with a First Pass
A straightforward way to do this would be to take the first element of our array, call that the root, and then iterate through the rest of our array, adding those elements as nodes in the binary search tree. In pseudocode, that might look something like this:
def create_min_height_bst(sorted_arr):
root = BinaryTreeNode(sorted_arr[0])
for elem in sorted_arr:
root.insert(elem)
return root
function createMinHeightBST(sortedArray) {
const left = 0;
const right = sortedArray.length - 1;
return recHelper(sortedArray, left, right);
}
function recHelper(sortedArray, left, right) {
if (left > right) {
return null;
}
const midpoint = math.floor(right - left) / 2 + left;
const root = new BinaryTreeNode(sortedArray[midpoint]);
root.left = recHelper(sortedArray, left, midpoint - 1);
root.right = recHelper(sortedArray, midpoint + 1, right);
return root;
}
class BinaryTreeNode {
constructor(value) {
this.value = value;
this.left = null;
this.right = null;
}
}
function isBST(root, minBound, maxBound) {
if (root === null) {
return true;
}
if (root.value < minBound || root.value > maxBound) {
return false;
}
const left = isBST(root.left, minBound, root.value - 1);
const right = isBST(root.right, root.value + 1, maxBound);
return left && right;
}
function findBSTMaxHeight(node) {
if (node === null) {
return 0;
}
return (
1 + Math.max(findBSTMaxHeight(node.left), findBSTMaxHeight(node.right))
);
}
function isBSTMinHeight(root, N) {
const height = findBSTMaxHeight(root);
const shouldEqual = Math.floor(Math.log2(N)) + 1;
return height === shouldEqual;
}
function countBSTNodes(root, count) {
if (root === null) {
return count;
}
countBSTNodes(root.left, count);
count++;
countBSTNodes(root.right, count);
}
// Some tests
let sortedArray = [1, 2, 3, 4, 5, 6, 7];
let bst = createMinHeightBST(sortedArray);
console.log(isBST(bst, -Infinity, Infinity));
console.log(isBSTMinHeight(bst, sortedArray.length));
sortedArray = [4, 10, 11, 18, 42, 43, 47, 49, 55, 67, 79, 89, 90, 95, 98, 100];
bst = createMinHeightBST(sortedArray);
console.log(isBST(bst, -Infinity, Infinity));
console.log(isBSTMinHeight(bst, sortedArray.length));
import math
def create_min_height_bst(sorted_array):
left = 0
right = len(sorted_array) - 1
return rec_helper(sorted_array, left, right)
def rec_helper(sorted_array, left, right):
if left > right:
return None
midpoint = ((right - left) // 2) + left
root = BinaryTreeNode(sorted_array[midpoint])
root.left = rec_helper(sorted_array, left, midpoint - 1)
root.right = rec_helper(sorted_array, midpoint + 1, right)
return root
class BinaryTreeNode:
def __init__(self, value):
self.value = value
self.left = None
self.right = None
# Helper function to validate that the created tree is a valid BST
def is_BST(root, min_bound, max_bound):
if root is None:
return True
if root.value < min_bound or root.value > max_bound:
return False
left = is_BST(root.left, min_bound, root.value - 1)
right = is_BST(root.right, root.value + 1, max_bound)
return left and right
# Helper function to check the max height of a BST
def find_bst_max_height(node):
if node is None:
return 0
return 1 + max(find_bst_max_height(node.left), find_bst_max_height(node.right))
# Helper function to validate that the given BST exhibits the min height
def is_bst_min_height(root, N):
bst_max_height = find_bst_max_height(root)
should_equal = math.floor(math.log2(N)) + 1
return bst_max_height == should_equal
# Helper function to count the number of nodes for a given BST
def count_bst_nodes(root, count):
if root is None:
return count
count_bst_nodes(root.left, count)
count += 1
count_bst_nodes(root.right, count)
# Some tests
sorted_array = [1, 2, 3, 4, 5, 6, 7]
bst = create_min_height_bst(sorted_array)
print(is_BST(bst, float("-inf"), float("inf"))) # should print true
print(is_bst_min_height(bst, len(sorted_array))) # should print true
sorted_array = [4, 10, 11, 18, 42, 43, 47, 49, 55, 67, 79, 89, 90, 95, 98, 100]
bst = create_min_height_bst(sorted_array)
print(is_BST(bst, float("-inf"), float("inf"))) # should print true
print(is_bst_min_height(bst, len(sorted_array))) # should print true
Another BST Implementation:
class BinarySearchTree:
def __init__(self, value):
self.value = value
self.left = None
self.right = None
def insert(self, value):
pass
def contains(self, target):
pass
def get_max(self):
pass
def for_each(self, cb):
pass
import unittest
import random
from binary_search_tree import BinarySearchTree
class BinarySearchTreeTests(unittest.TestCase):
def setUp(self):
self.bst = BinarySearchTree(5)
def test_insert(self):
self.bst.insert(2)
self.bst.insert(3)
self.bst.insert(7)
self.bst.insert(6)
self.assertEqual(self.bst.left.right.value, 3)
self.assertEqual(self.bst.right.left.value, 6)
def test_contains(self):
self.bst.insert(2)
self.bst.insert(3)
self.bst.insert(7)
self.assertTrue(self.bst.contains(7))
self.assertFalse(self.bst.contains(8))
def test_get_max(self):
self.assertEqual(self.bst.get_max(), 5)
self.bst.insert(30)
self.assertEqual(self.bst.get_max(), 30)
self.bst.insert(300)
self.bst.insert(3)
self.assertEqual(self.bst.get_max(), 300)
def test_for_each(self):
arr = []
cb = lambda x: arr.append(x)
v1 = random.randint(1, 101)
v2 = random.randint(1, 101)
v3 = random.randint(1, 101)
v4 = random.randint(1, 101)
v5 = random.randint(1, 101)
self.bst.insert(v1)
self.bst.insert(v2)
self.bst.insert(v3)
self.bst.insert(v4)
self.bst.insert(v5)
self.bst.for_each(cb)
self.assertTrue(5 in arr)
self.assertTrue(v1 in arr)
self.assertTrue(v2 in arr)
self.assertTrue(v3 in arr)
self.assertTrue(v4 in arr)
self.assertTrue(v5 in arr)
if __name__ == '__main__':
unittest.main()
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