Binary Search Tree | Set 1 (Search and Insertion)

What is Binary Search Tree?

A 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.

The above properties of the 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 order, then we may have to compare every key to search for a given key.

How to search a key in given Binary Tree?

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, in binary search, 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 middle element (median) of the search space and if the record being searched is less than the middle element, 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 if the mid element is not the element that we are looking for, we reduce the search space to ‘n/2’ we keep reducing the search space until 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 operations in binary search trees will be very similar. Let’s say we want to search for the number, we start at the root, and then we 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 smaller 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 smaller and all the elements in the right subtree are larger. Searching an element in the binary search tree is basically this traversal, at each step we go either left or right and 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 start with a search space of ‘n’ nodes and as we discard one of the sub-trees, we discard ‘n/2’ nodes so our search space gets reduced to ‘n/2’. In the next step, we reduce the search space to ‘n/4’ and we repeat until we find the element or our search space is reduced to only one node. The search here is also a binary search hence the name; Binary Search Tree.

Implementation:

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>

Time complexity: O(h), where h is the height of the BST.
Space complexity: O(h), where h is the height of the BST. This is because the maximum amount of space needed to store the recursion stack would be h.

Illustration to search 6 in below tree:

1. Start from the root. 
2. Compare the searching element with root, if less than root, then recursively call left subtree, else recursively call right subtree. 
3. If the element to search is found anywhere, return true, else return false. 
   

Insertion of a key :

A new key is always inserted at the leaf. We start searching for 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

Implementation:

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 if (value < root->data){
        // Insert left node data, if the 'value'
        // to be inserted is smaller 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
import java.io.*;
// 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; }

    BinarySearchTree(int value) { root = new Node(value); }

    // 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 */
        else 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; }

    BinarySearchTree(int value) { root = new Node(value); }

    // 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

Javascript

<script>
// javascript program to demonstrate
// insert operation in binary
// search tree
    /*
     * Class containing left and right child of current node and key value
     */
    class Node {

constructor(item) {
            this.key = item;
            this.left = this.right = null;
        }
    }

    // Root of BST
    var root = null;

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

    /*
     * A recursive function to insert a new key in BST
     */
    function insertRec(root , 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()
    function inorder() {
        inorderRec(root);
    }

    // A utility function to
    // do inorder traversal of BST
    function inorderRec(root)
    {
        if (root != null) {
            inorderRec(root.left);
            document.write(root.key+"<br/>");
            inorderRec(root.right);
        }
    }

// Driver Code

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

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

// This code is contributed by Rajput-Ji
</script>

Output

20
30
40
50
60
70
80

Illustration to insert 2 in the below tree:

1.  Start from the root. 
2. Compare the inserting element with the root, if less than the root, then recursively call the left subtree, else recursively call the right subtree. 
3. After reaching the end, just insert that node at left(if less than current) or else right. 
   

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 the root to the deepest leaf node. The height of a skewed tree may become n and the time complexity of the search and insert operation may become O(n). 
Auxiliary Space: O(1)

Implementation: Insertion using the loop.

C++

// C++ Code to insert node and to print inorder traversal
// using iteration
#include <bits/stdc++.h>
using namespace std;

// BST Node
class Node {
public:
    int val;
    Node* left;
    Node* right;
    Node(int val)
        : val(val)
        , left(NULL)
        , right(NULL)
    {
    }
};

// Utility function to insert node in BST
void insert(Node*& root, int key)
{
    Node* node = new Node(key);
    if (!root) {
        root = node;
        return;
    }
    Node* prev = NULL;
    Node* temp = root;
    while (temp) {
        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;
}

// Utility function to print inorder traversal
void inorder(Node* root)
{
    Node* temp = root;
    stack<Node*> st;
    while (temp != NULL || !st.empty()) {
        if (temp != NULL) {
            st.push(temp);
            temp = temp->left;
        }
        else {
            temp = st.top();
            st.pop();
            cout << temp->val << " ";
            temp = temp->right;
        }
    }
}

// Driver code
int main()
{
    Node* root = NULL;
    insert(root, 30);
    insert(root, 50);
    insert(root, 15);
    insert(root, 20);
    insert(root, 10);
    insert(root, 40);
    insert(root, 60);

    inorder(root);

    return 0;
}

// This code is contributed by Tapesh(tapeshdua420)

Java

import java.io.*;
import java.util.*;

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;
            }
        }
    }
}

Python3

class GFG:
    @staticmethod
    def main(args):
        tree = 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:
    left = None
    val = 0
    right = None

    def __init__(self, val):
        self.val = val


class BST:
    root = None

    def insert(self, key):
        node = Node(key)
        if (self.root == None):
            self.root = node
            return
        prev = None
        temp = self.root
        while (temp != None):
            if (temp.val > key):
                prev = temp
                temp = temp.left
            elif(temp.val < key):
                prev = temp
                temp = temp.right
        if (prev.val > key):
            prev.left = node
        else:
            prev.right = node

    def inorder(self):
        temp = self.root
        stack = []
        while (temp != None or not (len(stack) == 0)):
            if (temp != None):
                stack.append(temp)
                temp = temp.left
            else:
                temp = stack.pop()
                print(str(temp.val) + " ", end="")
                temp = temp.right


if __name__ == "__main__":
    GFG.main([])

    # This code is contributed by rastogik346.

C#

using System;
using System.Collections.Generic;

public 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();
    }
}

public class Node {
    public Node left;
    public int val;
    public Node right;

    public Node(int val) { this.val = val; }
}

public class BST {
    public 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<Node>();
        while (temp != null || stack.Count != 0) {
            if (temp != null) {
                stack.Push(temp);
                temp = temp.left;
            }
            else {
                temp = stack.Pop();
                Console.Write(temp.val + " ");
                temp = temp.right;
            }
        }
    }
}

// This code is contributed by Rajput-Ji

Javascript

// JS Code to insert node and to print inorder traversal
// using iteration

// BST Node
class Node {
    constructor(val) {
        this.val = val;
        this.right = null
        this.left = null
    }
}
// Utility function to insert node in BST
function insert(root, key) {
    let node = new Node(key);
    if (!root) {
        root = node;
        return;
    }
    let prev = null;
    let temp = root;
    while (temp) {
        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;
}
// Utility function to print inorder traversal
function inorder(root)
{
    let temp = root;
    let st=[];
    while (temp != null || st.length!=0) {
        if (temp != null) {
            st.unshift(temp);
            temp = temp.left;
        }
        else {
            temp = st[0];
            st.shift();
            if(temp.val){
                console.log(temp.val);
            }
            temp = temp.right;
        }
    }
}


// Driver code
let root = new Node(null);
insert(root, 30);
insert(root, 50);
insert(root, 15);
insert(root, 20);
insert(root, 10);
insert(root, 40);
insert(root, 60);

inorder(root);

// This code is contributed by adityamaharshi21

Output

10 15 20 30 40 50 60 

Time Complexity: O(N)
Auxiliary Space: O(N)

Some Interesting Facts:

• Inorder traversal of BST always produces sorted output.
• We can construct a BST with only Preorder or Postorder or Level Order traversal. Note that we can always get inorder traversal by sorting the only given traversal.
• Number of unique BSTs with n distinct keys is Catalan Number

Related Links:

• Binary Search Tree Delete Operation
• Quiz on Binary Search Tree
• Coding practice on BST
• All Articles on BST