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Binary Search Tree | Set 1 (Search and Insertion)

  • Difficulty Level : Easy
  • Last Updated : 04 Oct, 2021
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The following is the 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 lesser 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.

200px-Binary_search_tree.svg

The above properties of Binary Search Tree provides 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 for a given key.

 

Searching a key 
For searching a value, if we had a sorted array we could have performed a binary search. Let’s say we want to search a number in the array what we do in binary search is we first define the complete list as our search space, the number can exist only within the search space. Now we compare the number to be searched or the element to be searched with the mid element of the search space or the median and if the record being searched is lesser we go searching in the left half else we go searching in the right half, in case of equality we have found the element. In binary search we start with ‘n’ elements in search space and then if the mid element is not the element that we are looking for, we reduce the search space to ‘n/2’ and we go on reducing the search space till we either find the record that we are looking for or we get to only one element in search space and be done with this whole reduction. 

Search operation in binary search tree will be very similar. Let’s say we want to search for the number, what we’ll do is we’ll start at the root, and then we will compare the value to be searched with the value of the root if it’s equal we are done with the search if it’s lesser we know that we need to go to the left subtree because in a binary search tree all the elements in the left subtree are lesser and all the elements in the right subtree are greater. Searching an element in the binary search tree is basically this traversal in which at each step we will go either towards left or right and hence in at each step we discard one of the sub-trees. If the tree is balanced, we call a tree balanced if for all nodes the difference between the heights of left and right subtrees is not greater than one, we will start with a search space of ‘n’nodes and when we will discard one of the sub-trees we will discard ‘n/2’ nodes so our search space will be reduced to ‘n/2’ and then in the next step we will reduce the search space to ‘n/4’ and we will go on reducing like this till we find the element or till our search space is reduced to only one node. The search here is also a binary search and that’s why the name binary search tree.



C++




// 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);
}

Java




// 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;
 
    // 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);
}

Python




# 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

C#




// 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;
 
   // 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);
}
 
// This code is contributed by gauravrajput1

Javascript




<script>
 
// A utility function to search
// a given key in BST
function search(root, 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);
}
 
// This code is contributed by rrrtnx.
</script>

Illustration to search 6 in below tree: 
1. Start from the root. 
2. Compare the searching element with root, if less than root, then recurse for left, else recurse for right. 
3. If the element to search is found anywhere, return true, else return false. 
 

bstsearch

 
Insertion of a key 
A new key is always inserted at the leaf. We start searching a key from the root until 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++




// C++ program to demonstrate insertion
// in a BST recursively.
#include <iostream>
using namespace std;
 
class BST
{
    int data;
    BST *left, *right;
 
public:
    // Default constructor.
    BST();
 
    // Parameterized constructor.
    BST(int);
 
    // Insert function.
    BST* Insert(BST*, int);
 
    // Inorder traversal.
    void Inorder(BST*);
};
 
// Default Constructor definition.
BST ::BST()
    : data(0)
    , left(NULL)
    , right(NULL)
{
}
 
// Parameterized Constructor definition.
BST ::BST(int value)
{
    data = value;
    left = right = NULL;
}
 
// Insert function definition.
BST* BST ::Insert(BST* root, int value)
{
    if (!root)
    {
        // Insert the first node, if root is NULL.
        return new BST(value);
    }
 
    // Insert data.
    if (value > root->data)
    {
        // Insert right node data, if the 'value'
        // to be inserted is greater than 'root' node data.
 
        // Process right nodes.
        root->right = Insert(root->right, value);
    }
    else
    {
        // Insert left node data, if the 'value'
        // to be inserted is greater than 'root' node data.
 
        // Process left nodes.
        root->left = Insert(root->left, value);
    }
 
    // Return 'root' node, after insertion.
    return root;
}
 
// Inorder traversal function.
// This gives data in sorted order.
void BST ::Inorder(BST* root)
{
    if (!root) {
        return;
    }
    Inorder(root->left);
    cout << root->data << endl;
    Inorder(root->right);
}
 
// Driver code
int main()
{
    BST b, *root = NULL;
    root = b.Insert(root, 50);
    b.Insert(root, 30);
    b.Insert(root, 20);
    b.Insert(root, 40);
    b.Insert(root, 70);
    b.Insert(root, 60);
    b.Insert(root, 80);
 
    b.Inorder(root);
    return 0;
}
 
// This code is contributed by pkthapa

C




// 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 Code
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;
}

Java




// 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 Code
    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

Python




# 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, key):
    if root is None:
        return Node(key)
    else:
        if root.val == key:
            return root
        elif root.val < key:
            root.right = insert(root.right, key)
        else:
            root.left = insert(root.left, key)
    return root
 
# 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)
r = insert(r, 30)
r = insert(r, 20)
r = insert(r, 40)
r = insert(r, 70)
r = insert(r, 60)
r = insert(r, 80)
 
# Print inoder traversal of the BST
inorder(r)

C#




// C# program to demonstrate
// insert operation in binary
// search tree
using System;
 
class BinarySearchTree{
     
// Class containing left and
// right child of current node
// and key value
public class Node
{
    public int key;
    public 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);
        Console.WriteLine(root.key);
        inorderRec(root.right);
    }
}
 
// Driver Code
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 aashish1995
Output
20
30
40
50
60
70
80

Illustration to insert 2 in below tree: 
1. Start from the root. 
2. Compare the inserting element with root, if less than root, then recurse for left, else recurse for right. 
3. After reaching the end, just insert that node at left(if less than current) else right. 
 

bstsearch

Time Complexity: The worst-case time complexity of search and insert operations is O(h) where h is the height of the Binary Search Tree. In the 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). 

Insertion using loop:

Java




import java.util.*;
import java.io.*;
 
class GFG {
    public static void main (String[] args) {
         BST tree=new BST();
        tree.insert(30);
        tree.insert(50);
        tree.insert(15);
        tree.insert(20);
        tree.insert(10);
        tree.insert(40);
        tree.insert(60);
        tree.inorder();
    }
}
 
class Node{
    Node left;
    int val;
    Node right;
    Node(int val){
        this.val=val;
    }
}
 
class BST{
  Node root;
   
  public void insert(int key){
        Node node=new Node(key);
        if(root==null) {
            root = node;
            return;
        }
        Node prev=null;
        Node temp=root;
        while (temp!=null){
            if(temp.val>key){
                prev=temp;
                temp=temp.left;
            }
            else if (temp.val<key){
                prev=temp;
                temp=temp.right;
            }
        }
        if(prev.val>key)
            prev.left=node;
        else prev.right=node;
    }
   
   public void inorder(){
        Node temp=root;
        Stack<Node> stack=new Stack<>();
        while (temp!=null||!stack.isEmpty()){
            if(temp!=null){
                stack.add(temp);
                temp=temp.left;
            }
            else {
                temp=stack.pop();
                System.out.print(temp.val+" ");
                temp=temp.right;
            }
        }
    }
}
Output
10 15 20 30 40 50 60 
 

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