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C Binary Search Tree

Summary : this tutorial introduces you to binary search tree data structure and how to implement it in C.

Introduction to binary search tree

A binary search tree or BST is a binary tree in symmetric order.

A binary search tree can:

• Have a key and not more than two other subtrees, which are called left subtree and right subtree.

A binary search tree is in symmetric order, it means:

• Each node contains a key.
• Each node’s key is smaller than all node’s keys in the right subtree and bigger than all node’s keys in the left subtree.

A binary search tree is also known as sorted or ordered binary tree .

Binary search tree operations

There are some common operations on the binary search tree:

• Insert – inserts a new node into the tree
• Delete – removes an existing node from the tree
• Traverse – traverse the tree in pre-order, in-order and post-order. For the binary search tree, only in-order traversal makes sense
• Search – search for a given node’s key in the tree

All binary search tree operations are O(H), where H is the depth of the tree. The minimum height of a binary search tree is H = log 2 N, where N is the number of the tree’s nodes. Therefore the complexity of a binary search tree operation in the best case is O(logN); and in the worst case, its complexity is O(N).

The worst case happens when the binary search tree is unbalanced. Many algorithms have been invented to keep a binary search tree balanced such as the height-balanced tree or AVL trees of A delson- V elskii and L andis, B-trees, and Splay trees.

C binary search tree implementation

We can use a structure to model the binary search tree node a follows:

The node structure has three members:

• data : stores node’s key, which is an integer . You can store a string , a pointer to a structure or (void*) . For the sake of simplicity, we will use a binary search tree of integers.
• left : points to the left subtree.
• right : points to the right subtree.

To create a new node, we use the malloc() function to allocate memory dynamically as follows:

To compare the data of two nodes, we can compare two integers. However, to make it more extensible, we can pass a callback to any function that has a comparison between two keys of nodes. The callback for comparing two nodes is defined as follows:

Later on, you may change the type of data to string , float , or even void* , and change the comparer callback accordingly.

Insert a new node into the binary search tree

If the binary search tree is empty, we just create a new node, otherwise, we start from the root node:

We find the proper position for the new node by comparing the new node’s key with the root node’s key. If the key of the new node less than the root node’s key, we go to the left subtree. If the key of the new node is greater than the root node’s key, we go to the right subtree. We do this step recursively until we find the correct position in the tree to insert the new node.

The following illustrates the insert_node function:

Delete a node from the binary search tree

Deleting an existing node in the binary search tree is little more complicated. There are three cases that we should consider:

Case 1. Delete a leaf node i.e., the node that has no children. We just need to remove it. The following example illustrates how to remove the leaf node e.g., 13

Case 2. Remove a node that has 1 child node, we replace it with its child node and remove it e.g., to remove node 10 in the following picture.

Case 3. To remove a node that has two child nodes or two children, we find its in-order successor node, which is the next node in an in-order traversal of the tree, and replaces it with the in-order success node.

For example, to remove the node 3 , we find its in-order successor node, which is 4 , and replace the node 3 by 4 as illustrated in the following picture.

The following is the delete node function that uses the recursive function in C :

Search for a specific key in the binary search tree

To search for a specific key in the binary search tree, we start from the root node. If the tree is empty, the key does not exist. Otherwise, if the root node’s key is equal to the key, the search is successful, we terminate the search. If the key is less than the root node’s key, we search the left subtree. Likewise, if the key is greater than the root node’s key, we search the right subtree. We repeat this step recursively until the key is found or subtree is NULL.

Traverse the binary search tree

We can traverse the binary search tree once it is created by traversing recursively to the left subtree of the root node, accessing the root node’s data, and then recursively traversing to the right subtree of the root node. This is called in-order traversal of a binary tree.

Because of the rules of the nodes’ keys in the binary search tree, this in-order traversal always creates a sorted list of nodes.

The following is the in-order traversal function of the binary search tree:

Notice that we pass a callback to the traverse function so that we can manipulate the node dynamically. The callback is defined as follows:

Remove all nodes

We must deallocate memory allocated for all the nodes in the binary search tree. We can create a function named dispose() to do this:

C binary search tree program

Before creating a program to test our binary search tree, we need to create some functions for comparing two integers:

Displaying a node:

And a function for displaying the whole tree:

Now it’s time to play with the binary search tree:

The following is the output of the binary search tree program in C:

In this tutorial, you have learned how to implement the binary search tree in C.

Binary Search Trees: BST Explained with Examples

What is a Binary Search Tree?

A tree is a data structure composed of nodes that has the following characteristics:

• Each tree has a root node at the top (also known as Parent Node) containing some value (can be any datatype).
• The root node has zero or more child nodes.
• Each child node has zero or more child nodes, and so on. This creates a subtree in the tree. Every node has its own subtree made up of its children and their children, etc. This means that every node on its own can be a tree.

A binary search tree (BST) adds these two characteristics:

• Each node has a maximum of up to two children.
• For each node, the values of its left descendent nodes are less than that of the current node, which in turn is less than the right descendent nodes (if any).

The BST is built on the idea of the binary search algorithm, which allows for fast lookup, insertion and removal of nodes. The way that they are set up means that, on average, each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the logarithm of the number of items stored in the tree,   O(log n) . However, some times the worst case can happen, when the tree isn't balanced and the time complexity is   O(n)  for all three of these functions. That is why self-balancing trees (AVL, red-black, etc.) are a lot more effective than the basic BST.

Worst case scenario example:  This can happen when you keep adding nodes that are   always  larger than the node before (its parent), the same can happen when you always add nodes with values lower than their parents.

Basic operations on a BST

• Create: creates an empty tree.
• Insert: insert a node in the tree.
• Search: Searches for a node in the tree.
• Delete: deletes a node from the tree.
• Inorder: in-order traversal of the tree.
• Preorder: pre-order traversal of the tree.
• Postorder: post-order traversal of the tree.

Initially an empty tree without any nodes is created. The variable/identifier which must point to the root node is initialized with a   NULL  value.

You always start searching the tree at the root node and go down from there. You compare the data in each node with the one you are looking for. If the compared node doesn't match then you either proceed to the right child or the left child, which depends on the outcome of the following comparison: If the node that you are searching for is lower than the one you were comparing it with, you proceed to the left child, otherwise (if it's larger) you go to the right child. Why? Because the BST is structured (as per its definition), that the right child is always larger than the parent and the left child is always lesser.

Breadth-first search (BFS)

Breadth first search is an algorithm used to traverse a BST. It begins at the root node and travels in a lateral manner (side to side), searching for the desired node. This type of search can be described as O(n) given that each node is visited once and the size of the tree directly correlates to the length of the search.

Depth-first search (DFS)

With a Depth-first search approach, we start with the root node and travel down a single branch. If the desired node is found along that branch, great, but if not, continue upwards and search unvisited nodes. This type of search also has a big O notation of O(n).

It is very similar to the search function. You again start at the root of the tree and go down recursively, searching for the right place to insert our new node, in the same way as explained in the search function. If a node with the same value is already in the tree, you can choose to either insert the duplicate or not. Some trees allow duplicates, some don't. It depends on the certain implementation.

There are 3 cases that can happen when you are trying to delete a node. If it has,

• No subtree (no children): This one is the easiest one. You can simply just delete the node, without any additional actions required.
• One subtree (one child): You have to make sure that after the node is deleted, its child is then connected to the deleted node's parent.
• Two subtrees (two children): You have to find and replace the node you want to delete with its inorder successor (the leftmost node in the right subtree).

The time complexity for creating a tree is   O(1) . The time complexity for searching, inserting or deleting a node depends on the height of the tree   h , so the worst case is   O(h)  in case of skewed trees.

Predecessor of a node

Predecessors can be described as the node that would come right before the node you are currently at. To find the predecessor of the current node, look at the right-most/largest leaf node in the left subtree.

Successor of a node

Successors can be described as the node that would come right after the the current node. To find the successor of the current node, look at the left-most/smallest leaf node in the right subtree.

Special types of BT

• Red-black tree
• Trie (Radix tree)

Data structure: BST

• Worst-case performance:   O(n)
• Best-case performance:   O(1)
• Average performance:   O(log n)
• Worst-case space complexity:   O(1)

Where   n  is the number of nodes in the BST. Worst case is O(n) since BST can be unbalanced.

Implementation of BST

Here's a definition for a BST node having some data, referencing to its left and right child nodes.

Search Operation

Whenever an element is to be searched, start searching from the root node. Then if the data is less than the key value, search for the element in the left subtree. Otherwise, search for the element in the right subtree. Follow the same algorithm for each node.

Insert Operation

Whenever an element is to be inserted, first locate its proper location. Start searching from the root node, then if the data is less than the key value, search for the empty location in the left subtree and insert the data. Otherwise, search for the empty location in the right subtree and insert the data.

Delete Operation

Binary search trees (BSTs) also give us quick access to predecessors and successors. Predecessors can be described as the node that would come right before the node you are currently at.

• To find the predecessor of the current node, look at the rightmost/largest leaf node in the left subtree. Successors can be described as the node that would come right after the node you are currently at.
• To find the successor of the current node, look at the leftmost/smallest leaf node in the right subtree.

Let's look at a couple of procedures operating on trees.

Since trees are recursively defined, it's very common to write routines that operate on trees that are themselves recursive.

So for instance, if we want to calculate the height of a tree, that is the height of a root node, We can go ahead and recursively do that, going through the tree. So we can say:

• For instance, if we have a nil tree, then its height is a 0.
• Otherwise, We're 1 plus the maximum of the left child tree and the right child tree.
• So if we look at a leaf for example, that height would be 1 because the height of the left child is nil, is 0, and the height of the nil right child is also 0. So the max of that is 0, then 1 plus 0.

Height(tree) algorithm

Here is the code in c++.

We could also look at calculating the size of a tree that is the number of nodes.

• Again, if we have a nil tree, we have zero nodes.
• Otherwise, we have the number of nodes in the left child plus 1 for ourselves plus the number of nodes in the right child. So 1 plus the size of the left tree plus the size of the right tree.

Size(tree) algorithm

There are 3 kinds of traversals that are done typically over a binary search tree. All these traversals have a somewhat common way of going over the nodes of the tree.

This traversal first goes over the left subtree of the root node, then accesses the current node, followed by the right subtree of the current node. The code represents the base case too, which says that an empty tree is also a binary search tree.

This traversal first accesses the current node value, then traverses the left and right sub-trees respectively.

This traversal puts the root value at last, and goes over the left and right sub-trees first. The relative order of the left and right sub-trees remain the same. Only the position of the root changes in all the above mentioned traversals.

Relevant videos on freeCodeCamp YouTube channel

And Binary Search Tree: Traversal and Height

Following are common types of Binary Trees:

Full Binary Tree/Strict Binary Tree: A Binary Tree is full or strict if every node has exactly 0 or 2 children.

In Full Binary Tree, number of leaf nodes is equal to number of internal nodes plus one.

Complete Binary Tree: A Binary Tree is complete Binary Tree if all levels are completely filled except possibly the last level and the last level has all keys as left as possible

Perfect Binary Tree A Binary tree is Perfect Binary Tree in which all internal nodes have two children and all leaves are at the same level.

Augmenting a BST

Sometimes we need to store some additional information with the traditional data structures to make our tasks easier. For example, consider a scenario where you are supposed to find the ith smallest number in a set. You can use brute force here but we can reduce the complexity of the problem to O(lg n) by augmenting a red-black or any self-balancing tree (where n is the number of elements in the set). We can also compute rank of any element in O(lg n) time. Let us consider a case where we are augmenting a red-black tree to store the additional information needed. Besides the usual attributes, we can store number of internal nodes in the subtree rooted at x(size of the subtree rooted at x including the node itself). Let x be any arbitrary node of a tree.

x.size = x.left.size + x.right.size + 1

While augmenting the tree, we should keep in mind, that we should be able to maintain the augmented information as well as do other operations like insertion, deletion, updating in O(lg n) time.

Since, we know that the value of x.left.size will give us the number of nodes which proceed x in the order traversal of the tree. Thus, x.left.size + 1 is the rank of x within the subtree rooted at x.

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Binary Search Tree (BST)

In this tutorial, you will learn what is a binary search tree, how different operations like insertion, deletion, searching are done in a binary search tree with examples in C and what are the applications of binary search trees.

A Binary Search Tree is a special binary tree used for the efficient storage of data. The nodes in a binary search tree are arranged in order. It also allows for the simple insertion and deletion of items.

• It is a binary tree where each node can have a maximum of two childern.
• It is called a search tree, since it can search for and find an element with an avergae running time f(n) = O(log 2 n)

A binary search tree has the following properties:

• All the elements in the left subtree is less than the root node.
• All the elements in the right subtree is greater than the root node.
• These properties are shared by both the left and right trees i.e. they are also BSTs.

An example for a binary search tree is given below:

Operations on a Binary Search Tree

A binary search tree can perform three basic operations: searching, insertion, and deletion.

Searching in a Binary Search Tree

The search operation finds whether or not a particular value exists in a tree. Since the binary search tree is ordered, the search can be easily made.

Suppose we want to find X in a binary tree having root R.

• If X < R , then recursively search on the left subtree of the tree.
• If X > R , then recursively search on the right subtree of the tree.

Repeat these steps until we reach an empty subtree or locate a node R such that R = X . That is, we continue the search from the root down the tree until we reach X or a terminal node.

Algorithm: Search

Consider the following BST and suppose we want to find 43. Start from root node 50.

Move to the left subtree of root since 43<50 .

Now compare 43 and 41. Since 43>41 move to the right subtree.

Now 43<47 , So move to the right subtree, where 43 is found.

Inserting in a Binary Search Tree

The insert operation is similar to the search operation. This is because first, you have to find the correct position where the new element is to be inserted.

• If the tree is NULL, insert new element as the root.
• Otherwise, depending on the value, we continue to the right or left subtree, and when we reach a position where the left or right subtree is null, we insert the new node there.

Algorithm: Insert

Consider the following BST and suppose we want to insert a new node 42.

Start searching for element 42 from the root. The search will stop when we reach node 43.

Since 43>42 but 43 has no left subtree. So we insert 43 as the left child of node 43.

Deleting from a Binary Search Tree

The deletion operation is to be done very carefully that the properties of the binary search tree are not violated and no nodes are lost.

Deletion is done in the following three cases:

• Deleting a node that has no children.
• Deleting a node with one child.
• Deleting a node with two children.

Case 1: Deleting a node that has no children

Here the node we have to delete is a leaf node. So, we can simply delete the node from the tree.

Suppose we want to delete node 58 from the following tree.

Simply delete node 58.

Case 2: Deleting a node with one child.

The following steps are to be done to delete a node that has only a single child.

• Replace the node with its child.
• Remove the child node.

Suppose we want to delete node 47 from the following tree.

Replace node 47 with its child node 43.

Now delete the child node 43 from its original position

Case 3: Deleting a node with two children

The case is when the node we want to delete has 2 children. This case can be handled by the following steps:

• Replace the node’s value with it’s in-order successor.
• Remove the in-order successor from its original position.

Suppose we want to delete node 41 from the following tree.

Replace the value of node 41 with its in-order successor 43.

Now delete node 43 from its original position.

Algorithm: Deletion

Applications of Binary Search Tree

• Used for indexing and multilevel indexing in the database.
• Used to implement various searching algorithms.
• For dynamic sorting.
• Used for managing virtual memory areas in unix kernal.

Browse Course Material

Course info, instructors.

• Prof. Erik Demaine
• Prof. Srini Devadas

Departments

• Electrical Engineering and Computer Science

As Taught In

• Algorithms and Data Structures

Learning Resource Types

Introduction to algorithms, binary search trees.

This page contains various implementations of different Binary Search Trees (BSTs).

Simple BST (no balancing)

• Features: insert, find, delete_min, ASCII art
• Imports and extends bst.py
• Augmentation to compute subtree sizes
• Recursive version from recitation for computing subtree sizes
• Features: insert, find, rank, delete

Both bst.py and avl.py (as well as bstsize.py) can be tested interactively from a UNIX shell as follows:

• python bst.py 10 — do 10 random insertions, printing BST at each step
• python avl.py 10 — do 10 random insertions, printing AVL tree at each step

Alternatively, you can use them from a Python shell as follows:

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Python Operators

Precedence and associativity of operators in python.

• Python Arithmetic Operators
• Difference between / vs. // operator in Python
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• Division Operators in Python
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• Python Logical Operators
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• not Operator in Python | Boolean Logic

Ternary Operator in Python

• Python Bitwise Operators

Python Assignment Operators

Assignment operators in python.

• Walrus Operator in Python 3.8
• Increment += and Decrement -= Assignment Operators in Python
• Merging and Updating Dictionary Operators in Python 3.9
• New '=' Operator in Python3.8 f-string

Python Relational Operators

• Comparison Operators in Python
• Python NOT EQUAL operator
• Difference between == and is operator in Python
• Chaining comparison operators in Python
• Python Membership and Identity Operators
• Difference between != and is not operator in Python

In Python programming, Operators in general are used to perform operations on values and variables. These are standard symbols used for logical and arithmetic operations. In this article, we will look into different types of Python operators. 

• OPERATORS: These are the special symbols. Eg- + , * , /, etc.
• OPERAND: It is the value on which the operator is applied.

Types of Operators in Python

• Arithmetic Operators
• Comparison Operators
• Logical Operators
• Bitwise Operators
• Assignment Operators
• Identity Operators and Membership Operators

Arithmetic Operators in Python

Python Arithmetic operators are used to perform basic mathematical operations like addition, subtraction, multiplication , and division .

In Python 3.x the result of division is a floating-point while in Python 2.x division of 2 integers was an integer. To obtain an integer result in Python 3.x floored (// integer) is used.

Example of Arithmetic Operators in Python

Division operators.

In Python programming language Division Operators allow you to divide two numbers and return a quotient, i.e., the first number or number at the left is divided by the second number or number at the right and returns the quotient. 

There are two types of division operators: 

Float division

• Floor division

The quotient returned by this operator is always a float number, no matter if two numbers are integers. For example:

Example: The code performs division operations and prints the results. It demonstrates that both integer and floating-point divisions return accurate results. For example, ’10/2′ results in ‘5.0’ , and ‘-10/2’ results in ‘-5.0’ .

Integer division( Floor division)

The quotient returned by this operator is dependent on the argument being passed. If any of the numbers is float, it returns output in float. It is also known as Floor division because, if any number is negative, then the output will be floored. For example:

Example: The code demonstrates integer (floor) division operations using the // in Python operators . It provides results as follows: ’10//3′ equals ‘3’ , ‘-5//2’ equals ‘-3’ , ‘ 5.0//2′ equals ‘2.0’ , and ‘-5.0//2’ equals ‘-3.0’ . Integer division returns the largest integer less than or equal to the division result.

Precedence of Arithmetic Operators in Python

The precedence of Arithmetic Operators in Python is as follows:

• P – Parentheses
• E – Exponentiation
• M – Multiplication (Multiplication and division have the same precedence)
• D – Division
• A – Addition (Addition and subtraction have the same precedence)
• S – Subtraction

The modulus of Python operators helps us extract the last digit/s of a number. For example:

• x % 10 -> yields the last digit
• x % 100 -> yield last two digits

Arithmetic Operators With Addition, Subtraction, Multiplication, Modulo and Power

Here is an example showing how different Arithmetic Operators in Python work:

Example: The code performs basic arithmetic operations with the values of ‘a’ and ‘b’ . It adds (‘+’) , subtracts (‘-‘) , multiplies (‘*’) , computes the remainder (‘%’) , and raises a to the power of ‘b (**)’ . The results of these operations are printed.

Note: Refer to Differences between / and // for some interesting facts about these two Python operators.

Comparison of Python Operators

In Python Comparison of Relational operators compares the values. It either returns True or False according to the condition.

= is an assignment operator and == comparison operator.

Precedence of Comparison Operators in Python

In Python, the comparison operators have lower precedence than the arithmetic operators. All the operators within comparison operators have the same precedence order.

Example of Comparison Operators in Python

Let’s see an example of Comparison Operators in Python.

Example: The code compares the values of ‘a’ and ‘b’ using various comparison Python operators and prints the results. It checks if ‘a’ is greater than, less than, equal to, not equal to, greater than, or equal to, and less than or equal to ‘b’ .

Logical Operators in Python

Python Logical operators perform Logical AND , Logical OR , and Logical NOT operations. It is used to combine conditional statements.

Precedence of Logical Operators in Python

The precedence of Logical Operators in Python is as follows:

• Logical not
• logical and

Example of Logical Operators in Python

The following code shows how to implement Logical Operators in Python:

Example: The code performs logical operations with Boolean values. It checks if both ‘a’ and ‘b’ are true ( ‘and’ ), if at least one of them is true ( ‘or’ ), and negates the value of ‘a’ using ‘not’ . The results are printed accordingly.

Bitwise Operators in Python

Python Bitwise operators act on bits and perform bit-by-bit operations. These are used to operate on binary numbers.

Precedence of Bitwise Operators in Python

The precedence of Bitwise Operators in Python is as follows:

• Bitwise NOT
• Bitwise Shift
• Bitwise AND
• Bitwise XOR

Here is an example showing how Bitwise Operators in Python work:

Example: The code demonstrates various bitwise operations with the values of ‘a’ and ‘b’ . It performs bitwise AND (&) , OR (|) , NOT (~) , XOR (^) , right shift (>>) , and left shift (<<) operations and prints the results. These operations manipulate the binary representations of the numbers.

Python Assignment operators are used to assign values to the variables.

Let’s see an example of Assignment Operators in Python.

Example: The code starts with ‘a’ and ‘b’ both having the value 10. It then performs a series of operations: addition, subtraction, multiplication, and a left shift operation on ‘b’ . The results of each operation are printed, showing the impact of these operations on the value of ‘b’ .

Identity Operators in Python

In Python, is and is not are the identity operators both are used to check if two values are located on the same part of the memory. Two variables that are equal do not imply that they are identical. 

Example Identity Operators in Python

Let’s see an example of Identity Operators in Python.

Example: The code uses identity operators to compare variables in Python. It checks if ‘a’ is not the same object as ‘b’ (which is true because they have different values) and if ‘a’ is the same object as ‘c’ (which is true because ‘c’ was assigned the value of ‘a’ ).

Membership Operators in Python

In Python, in and not in are the membership operators that are used to test whether a value or variable is in a sequence.

Examples of Membership Operators in Python

The following code shows how to implement Membership Operators in Python:

Example: The code checks for the presence of values ‘x’ and ‘y’ in the list. It prints whether or not each value is present in the list. ‘x’ is not in the list, and ‘y’ is present, as indicated by the printed messages. The code uses the ‘in’ and ‘not in’ Python operators to perform these checks.

in Python, Ternary operators also known as conditional expressions are operators that evaluate something based on a condition being true or false. It was added to Python in version 2.5. 

It simply allows testing a condition in a single line replacing the multiline if-else making the code compact.

Syntax :   [on_true] if [expression] else [on_false] 

Examples of Ternary Operator in Python

The code assigns values to variables ‘a’ and ‘b’ (10 and 20, respectively). It then uses a conditional assignment to determine the smaller of the two values and assigns it to the variable ‘min’ . Finally, it prints the value of ‘min’ , which is 10 in this case.

In Python, Operator precedence and associativity determine the priorities of the operator.

Operator Precedence in Python

This is used in an expression with more than one operator with different precedence to determine which operation to perform first.

Let’s see an example of how Operator Precedence in Python works:

Example: The code first calculates and prints the value of the expression 10 + 20 * 30 , which is 610. Then, it checks a condition based on the values of the ‘name’ and ‘age’ variables. Since the name is “ Alex” and the condition is satisfied using the or operator, it prints “Hello! Welcome.”

Operator Associativity in Python

If an expression contains two or more operators with the same precedence then Operator Associativity is used to determine. It can either be Left to Right or from Right to Left.

The following code shows how Operator Associativity in Python works:

Example: The code showcases various mathematical operations. It calculates and prints the results of division and multiplication, addition and subtraction, subtraction within parentheses, and exponentiation. The code illustrates different mathematical calculations and their outcomes.

To try your knowledge of Python Operators, you can take out the quiz on Operators in Python . 

Python Operator Exercise Questions

Below are two Exercise Questions on Python Operators. We have covered arithmetic operators and comparison operators in these exercise questions. For more exercises on Python Operators visit the page mentioned below.

Q1. Code to implement basic arithmetic operations on integers

Q2. Code to implement Comparison operations on integers

Explore more Exercises: Practice Exercise on Operators in Python

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