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Data Structures

## Search and Insertion

The following is definition of Binary Search Tree(BST) according to Wikipedia

Binary Search Tree, is a node-based binary tree data structure which has the following properties:

• The left subtree of a node contains only nodes with keys less than the node’s key.
• The right subtree of a node contains only nodes with keys greater than the node’s key.
• The left and right subtree each must also be a binary search tree.
There must be no duplicate nodes.

The above properties of Binary Search Tree provide an ordering among keys so that the operations like search, minimum and maximum can be done fast. If there is no ordering, then we may have to compare every key to search a given key.

Searching a key
To search a given key in Bianry Search Tree, we first compare it with root, if the key is present at root, we return root. If key is greater than root’s key, we recur for right subtree of root node. Otherwise we recur for left subtree.

// C function to search a given key in a given BST
struct node* search(struct node* root, int key)
{
// Base Cases: root is null or key is present at root
if (root == NULL || root->key == key)
return root;

// Key is greater than root's key
if (root->key < key)
return search(root->right, key);

// Key is smaller than root's key
return search(root->left, key);
}


# A utility function to search a given key in BST
def search(root,key):

# Base Cases: root is null or key is present at root
if root is None or root.val == key:
return root

# Key is greater than root's key
if root.val < key:
return search(root.right,key)

# Key is smaller than root's key
return search(root.left,key)

# This code is contributed by Bhavya Jain


// A utility function to search a given key in BST
public Node search(Node root, int key)
{
// Base Cases: root is null or key is present at root
if (root==null || root.key==key)
return root;

// val is greater than root's key
if (root.key > key)
return search(root.left, key);

// val is less than root's key
return search(root.right, key);
}


Insertion of a key
A new key is always inserted at leaf. We start searching a key from root till we hit a leaf node. Once a leaf node is found, the new node is added as a child of the leaf node.


100                               100

/   \        Insert 40            /    \

20     500    --------->          20     500

/  \                              /  \

10   30                           10   30

\

40



// C program to demonstrate insert operation in binary search tree
#include<stdio.h>
#include<stdlib.h>

struct node
{
int key;
struct node *left, *right;
};

// A utility function to create a new BST node
struct node *newNode(int item)
{
struct node *temp =  (struct node *)malloc(sizeof(struct node));
temp->key = item;
temp->left = temp->right = NULL;
return temp;
}

// A utility function to do inorder traversal of BST
void inorder(struct node *root)
{
if (root != NULL)
{
inorder(root->left);
printf("%d \n", root->key);
inorder(root->right);
}
}

/* A utility function to insert a new node with given key in BST */
struct node* insert(struct node* node, int key)
{
/* If the tree is empty, return a new node */
if (node == NULL) return newNode(key);

/* Otherwise, recur down the tree */
if (key < node->key)
node->left  = insert(node->left, key);
else if (key > node->key)
node->right = insert(node->right, key);

/* return the (unchanged) node pointer */
return node;
}

// Driver Program to test above functions
int main()
{
/* Let us create following BST
50
/     \
30      70
/  \    /  \
20   40  60   80 */
struct node *root = NULL;
root = insert(root, 50);
insert(root, 30);
insert(root, 20);
insert(root, 40);
insert(root, 70);
insert(root, 60);
insert(root, 80);

// print inoder traversal of the BST
inorder(root);

return 0;
}


# Python program to demonstrate insert operation in binary search tree

# A utility class that represents an individual node in a BST
class Node:
def __init__(self,key):
self.left = None
self.right = None
self.val = key

# A utility function to insert a new node with the given key
def insert(root,node):
if root is None:
root = node
else:
if root.val < node.val:
if root.right is None:
root.right = node
else:
insert(root.right, node)
else:
if root.left is None:
root.left = node
else:
insert(root.left, node)

# A utility function to do inorder tree traversal
def inorder(root):
if root:
inorder(root.left)
print(root.val)
inorder(root.right)

# Driver program to test the above functions
# Let us create the following BST
#      50
#    /    \
#   30     70
#   / \    / \
#  20 40  60 80
r = Node(50)
insert(r,Node(30))
insert(r,Node(20))
insert(r,Node(40))
insert(r,Node(70))
insert(r,Node(60))
insert(r,Node(80))

# Print inoder traversal of the BST
inorder(r)

# This code is contributed by Bhavya Jain


// Java program to demonstrate insert operation in binary search tree
class BinarySearchTree {

/* Class containing left and right child of current node and key value*/
class Node {
int key;
Node left, right;

public Node(int item) {
key = item;
left = right = null;
}
}

// Root of BST
Node root;

// Constructor
BinarySearchTree() {
root = null;
}

// This method mainly calls insertRec()
void insert(int key) {
root = insertRec(root, key);
}

/* A recursive function to insert a new key in BST */
Node insertRec(Node root, int key) {

/* If the tree is empty, return a new node */
if (root == null) {
root = new Node(key);
return root;
}

/* Otherwise, recur down the tree */
if (key < root.key)
root.left = insertRec(root.left, key);
else if (key > root.key)
root.right = insertRec(root.right, key);

/* return the (unchanged) node pointer */
return root;
}

// This method mainly calls InorderRec()
void inorder()  {
inorderRec(root);
}

// A utility function to do inorder traversal of BST
void inorderRec(Node root) {
if (root != null) {
inorderRec(root.left);
System.out.println(root.key);
inorderRec(root.right);
}
}

// Driver Program to test above functions
public static void main(String[] args) {
BinarySearchTree tree = new BinarySearchTree();

/* Let us create following BST
50
/     \
30      70
/  \    /  \
20   40  60   80 */
tree.insert(50);
tree.insert(30);
tree.insert(20);
tree.insert(40);
tree.insert(70);
tree.insert(60);
tree.insert(80);

// print inorder traversal of the BST
tree.inorder();
}
}
// This code is contributed by Ankur Narain Verma


Output:

20
30
40
50
60
70
80
Time Complexity: The worst case time complexity of search and insert operations is O(h) where h is height of Binary Search Tree. In worst case, we may have to travel from root to the deepest leaf node. The height of a skewed tree may become n and the time complexity of search and insert operation may become O(n).

Quiz on Binary Search Tree