# DS-Binary Search Tree (BST) implementation in Python

6 min readMay 24, 2023

A binary search tree (BST) is a binary tree data structure in which each node has a key (or value) associated with it. The key in each node is greater than all keys in its left subtree and smaller than all keys in its right subtree.

# Difference between Tree, Binary Tree and Binary Search Tree.

## Tree:

- A hierarchical data structure composed of nodes.
- Consists of a collection of nodes where each node can have zero or more child nodes.
- No specific rules or constraints regarding the organization of the nodes.

## Binary Tree:

- A type of tree structure where each node has at most two children, referred to as the left child and the right child.
- Each node can have zero, one, or two children.
- The order or arrangement of the children is not significant.
- Useful for representing hierarchical relationships and recursive data structures.

## Binary Search Tree:

- A specific type of binary tree that satisfies an additional property or constraint.
- Each node in a binary search tree has a key value associated with it.
- The key value in each node is greater than all keys in its left subtree and smaller than all keys in its right subtree.
- Provides efficient search, insertion, and deletion operations for sorted data.
- Allows for efficient binary search by recursively traversing left or right subtrees based on the comparison of key values.

# Nodes properties of a BST

The nodes are arranged in a binary search tree according to the following properties:

- The left subtree of a particular node will always contain nodes with keys less than that node’s key.
- The right subtree of a particular node will always contain nodes with keys greater than that node’s key.
- The left and the right subtree of a particular node will also, in turn, be binary search trees.

## Time Complexity

- Average cases :
*O*(*logn*) - Worst case :
, when the tree is unbalanced.*O*(*n*) - Space Complexity :
**O(n)**

# Types of Traversals

- Pre-order Traversal
- In-order Traversal
- Post-order Traversal

# In-order Traversal

`class Node():`

def __init__(self, data):

self.left = None

self.right = None

self.data = data

def insert(node, value):

if node is None:

return Node(value)

if value < node.data:

node.left = insert(node.left, value)

def inOrderTraverse(node):

if node is not None:

inOrderTraverse(node.left)

print(node.data)

inOrderTraverse(node.right)

node = Node(5)

node.left = Node(3)

node.right = Node(7)

node.left.left = Node(2)

node.left.right = Node(4)

node.right.left = Node(6)

node.right.right = Node(8)

# output :

Inorder Traversal

2

3

4

5

6

7

8

# Pre-order Traversal

`class Node():`

def __init__(self, data):

self.left = None

self.right = None

self.data = data

def preOrderTraversal(node):

if node is not None:

print(node.data)

preOrderTraversal(node.left)

preOrderTraversal(node.right)

node = Node(5)

node.left = Node(3)

node.right = Node(7)

node.left.left = Node(2)

node.left.right = Node(4)

node.right.left = Node(6)

node.right.right = Node(8)

# output :

Preorder Traversal

5

3

2

4

7

6

8

# Post-order Traversal

`class Node():`

def __init__(self, data):

self.left = None

self.right = None

self.data = data

def postOrderTraversal(node):

if node is not None:

postOrderTraversal(node.left)

postOrderTraversal(node.right)

print(node.data)

node = Node(5)

node.left = Node(3)

node.right = Node(7)

node.left.left = Node(2)

node.left.right = Node(4)

node.right.left = Node(6)

node.right.right = Node(8)

# output :

Postorder Traversal

2

4

3

6

8

7

5

# Insert data into BST

`1. Compare the value with the current node.`

2. If the node is None, create a new node with the value and return it.

3. If the value is equal to the node's data, indicate that the value already exists and return.

4. If the value is less than the node's data, recursively insert it into the left subtree.

5. If the value is greater than the node's data, recursively insert it into the right subtree.

`class Node:`

def __init__(self, data):

self.data = data

self.left = None

self.right = None

def insert(node, value):

if node is None:

return Node(value)

if value == node.data:

print("value alredy exists")

return

elif value < node.data:

node.left = insert(node.left, value)

else:

if value > node.data:

node.right = insert(node.right, value)

return node

def printNode(node):

if node is not None:

printNode(node.left)

print(node.data)

printNode(node.right)

node = Node(5)

insert(node, 3)

insert(node, 7)

insert(node, 2)

insert(node, 4)

insert(node, 6)

insert(node, 8)

printNode(node)

# Output : 2,3,4,5,6,7,8

# Deleting Node from BST

## Case 1: Deleting a Leaf Node

- If the node to be deleted is a leaf node (i.e., it has no children), we can simply remove the node from the tree.

## Case 2: Deleting a Node with a Single Child

- If the node to be deleted has only one child, we can replace the node with its child. The child node is connected to the parent of the node to be deleted.

## Case 3: Deleting a Node with Two Children

- If the node to be deleted has two children, we need to find its successor or predecessor node.
- The successor node is the smallest node in the right subtree of the node to be deleted.
- The predecessor node is the largest node in the left subtree of the node to be deleted.
- We can replace the node to be deleted with its successor or predecessor node and recursively delete the successor or predecessor node from its original position.

`def delete(node, value):`

if node is None:

return node

if value < node.data:

node.left = delete(node.left, value)

elif value > node.data:

node.right = delete(node.right, value)

else:

# if node is a leaf node

if node.left is None and node.right is None:

node = None

# if root has one left child

elif node.left is None:

node = node.right

# if node has one right child

elif node.right is None:

node = node.left

# if node has both left and right child

else:

successor = findSuccessor(node.right)

node.data = successor.data

node.right = delete(node.right, successor.data)

return node

- The function takes a node and a value to be deleted from the binary search tree.
- The base case is checked first: if the node is None, indicating an empty subtree, it is returned as is.
- If the value is less than the node.data, the function recursively calls delete on the left subtree, updating the left child accordingly.
- If the value is greater than the node.data, the function recursively calls delete on the right subtree, updating the right child accordingly.
- If the value is equal to the node.data, indicating the node to be deleted, the following cases are considered:

- If the node is a leaf node (i.e., it has no left or right child), it is simply set to None.
- If the node has only a left child, the node is replaced by its left child.
- If the node has only a right child, the node is replaced by its right child.
- If the node has both a left and right child, the successor is found by locating the inorder successor in the right subtree. The node.data is updated with the successor’s data, and the successor node is deleted from the right subtree recursively.

6. Finally, the modified node is returned.