Introduction to Recursion

<!DOCTYPE html>    <html lang="en-US" prefix="og: http://ogp.me/ns#" >  <head> <meta charset="UTF-8" /> <meta name="keywords" content="Data Structures,Algorithms,Python,Java,C,C++,JavaScript,Android Development,SQL,Data Science,Machine Learning,PHP,Web Development,System Design,Tutorial,Technical Blogs,Interview Experience,Interview Preparation,Programming,Competitive Programming,Jobs,Coding Contests,GATE CSE,HTML,CSS,React,NodeJS,Placement,Aptitude,Quiz,Computer Science,Programming Examples,GeeksforGeeks Courses,Puzzles,SSC,Banking,UPSC,Commerce,Finance,CBSE,School,k12,General Knowledge,News,Mathematics,Exams" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, minimum-scale=0.5, maximum-scale=3.0"> <link rel="shortcut icon" href="https://media.geeksforgeeks.org/wp-content/cdn-uploads/gfg_favicon.png" type="image/x-icon" /> <link rel="preconnect" href="https://fonts.googleapis.com"> <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> <meta name="theme-color" content="#308D46" /> <meta name='robots' content='index, follow, max-image-preview:large, max-snippet:-1' /> <meta name="image" property="og:image" content="https://media.geeksforgeeks.org/wp-content/cdn-uploads/gfg_200x200-min.png"> <meta property="og:image:type" content="image/png"> <meta property="og:image:width" content="200"> <meta property="og:image:height" content="200"> <meta name="facebook-domain-verification" content="xo7t4ve2wn3ywfkjdvwbrk01pvdond" /> <script src="https://cdnads.geeksforgeeks.org/synchronously_gfg_ads.min.js"></script> <script defer src="https://apis.google.com/js/platform.js"></script> <script src="//cdnjs.cloudflare.com/ajax/libs/require.js/2.1.14/require.min.js"></script>     <script> var IHPWT={}; //Initialize Namespace var pbjs = pbjs || {}; pbjs.que = pbjs.que || []; var googletag = googletag || {}; googletag.cmd = googletag.cmd || []; var gptRan = false; </script> <script defer src="https://ads.pubmatic.com/AdServer/js/pwt/162080/12331/pwt.js"></script> <script defer src="https://securepubads.g.doubleclick.net/tag/js/gpt.js"></script> <script defer src="https://cdnads.geeksforgeeks.org/prebid.js?ver=0.1"></script> <script defer src="https://cdnads.geeksforgeeks.org/gfg_ads.min.js?ver=0.1"></script>  <script defer src="https://assets.geeksforgeeks.org/codetabs/v1/codetabs.js" environment="production" domain="gfg" no-scroll></script> <title>Introduction to Recursion - GeeksforGeeks</title> <link rel="profile" href="http://gmpg.org/xfn/11" /> <link rel="pingback" href="" />   <script> var video_api_response = JSON.parse("{\"id\":10855,\"title\":\"Introduction of Recursion\",\"slug\":\"introduction-of-recursion\",\"description\":\"<h2>Introduction of Recursion in Programming<\/strong><\/h2>In this video, we explore the concept of Recursion in Programming<\/strong>, a technique where a function calls itself in order to solve a problem. Recursion is a powerful and elegant approach to solving complex problems by breaking them down into smaller, more manageable sub-problems. In this tutorial, we\u2019ll cover the basics of recursion, its key principles, and how to implement it in your own programs.<\/p><h3>Key Features of Recursion:<\/h3><ul><li>Self-Referencing<\/strong>: A recursive function calls itself with modified parameters to work on smaller instances of the problem.<\/li><li>Base Case<\/strong>: Recursion relies on a base case to terminate the recursive calls and prevent infinite loops.<\/li><li>Simplification of Problems<\/strong>: Recursion is particularly effective for problems that can be naturally divided into smaller sub-problems, such as tree traversal, sorting algorithms, and more.<\/li><\/ul><h3>Steps to Implement Recursion:<\/h3><ul><li>Identify the Base Case<\/strong>: Determine the simplest instance of the problem that can be solved directly without further recursion.<\/li><li>Define the Recursive Case<\/strong>: Define how the function should break the problem into smaller sub-problems and call itself with those smaller instances.<\/li><li>Ensure Termination<\/strong>: Ensure that each recursive call progresses toward the base case to avoid infinite recursion.<\/li><\/ul><h3>Common Mistakes to Avoid:<\/h3><ul><li>Missing Base Case<\/strong>: Failing to include a base case is a common error that can lead to infinite recursion and stack overflow.<\/li><li>Excessive Recursion Depth<\/strong>: Recursive solutions can sometimes exceed the maximum recursion depth, leading to a stack overflow. Be mindful of the problem size.<\/li><li>Not Breaking Down the Problem<\/strong>: Ensure that each recursive step makes progress toward solving a smaller instance of the problem.<\/li><\/ul><h3>Applications of Recursion:<\/h3><ul><li>Tree Traversal<\/strong>: Recursion is widely used in traversing data structures like trees and graphs.<\/li><li>Sorting Algorithms<\/strong>: Algorithms like Merge Sort and Quick Sort use recursion to break down the problem into smaller sub-problems.<\/li><li>Dynamic Programming<\/strong>: Many dynamic programming problems are solved using recursion to build up solutions from smaller sub-problems.<\/li><li>Backtracking Algorithms<\/strong>: Problems like Sudoku, N-Queens, and other puzzle-solving algorithms are often solved using recursion.<\/li><\/ul><h3>Why Learn Recursion?<\/h3>Understanding recursion is crucial for developing an in-depth understanding of algorithms and data structures. By mastering recursion, you will:<\/p><ul><li>Enhance Problem-Solving Skills<\/strong>: Learn how to solve complex problems by breaking them down into smaller, manageable sub-problems.<\/li><li>Boost Algorithmic Thinking<\/strong>: Recursion is a core concept used in many classic algorithms, and understanding it will significantly improve your ability to tackle algorithmic challenges.<\/li><li>Deepen Your Programming Knowledge<\/strong>: Recursion is widely used in various programming paradigms, and knowing it will allow you to write more efficient and elegant code.<\/li><\/ul><h3>Topics Covered:<\/h3><ul><li>Basics of Recursion<\/strong>: Understand the fundamental concepts of recursion and its components.<\/li><li>Recursive Functions<\/strong>: Learn how to implement recursive functions in different programming languages.<\/li><li>Real-World Examples<\/strong>: Discover real-world applications of recursion in solving problems like tree traversal and sorting.<\/li><\/ul>\",\"source\":\"https:\/\/cdnvideos.geeksforgeeks.org\/5960745ad90233da237d3065a9761af6gfg-e9b33f56dbf200c31dff996e0b50013bgfgRecursion20241106124909\/video.m3u8\",\"category\":[{\"term_id__id\":125,\"term_id__term_name\":\"Data Structures\",\"term_id__term_type\":2,\"term_id__slug\":\"data-structures-bczc7q\"},{\"term_id__id\":365,\"term_id__term_name\":\"Data Structure and Algorithm\",\"term_id__term_type\":1,\"term_id__slug\":\"data-structure-and-algorithm\"},{\"term_id__id\":562,\"term_id__term_name\":\"DSA\",\"term_id__term_type\":1,\"term_id__slug\":\"dsa-lpubwc\"}],\"video_type\":\"courses\",\"meta\":{\"thumbnail\":\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804-small.png\",\"largeThumbnail\":\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804.jpg\",\"likes\":0,\"views\":40,\"isFeatured\":0,\"isPremium\":0,\"isPublic\":1,\"format\":\"video\/mp4\",\"revision\":{}},\"time\":\"06\/11\/2024\",\"seo\":{\"description\":\"<h2>Introduction of Recursion in Programming<\/strong><\/h2>In thi\",\"thumbnail\":\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804-seo.png\"},\"subtitle\":\"https:\/\/cdnvideos.geeksforgeeks.org\/5960745ad90233da237d3065a9761af6gfg-e9b33f56dbf200c31dff996e0b50013bgfgRecursion20241106124909\/subtitle.vtt\",\"duration\":621,\"post_list\":[{\"post_id\":140498,\"post_title\":\"Introduction to Recursion\"},{\"post_id\":449297,\"post_title\":\"Recursion in Python\"},{\"post_id\":1028894,\"post_title\":\"Recursive Algorithms\"},{\"post_id\":977298,\"post_title\":\"What is Recursion?\"},{\"post_id\":259388,\"post_title\":\"Difference between Recursion and Iteration\"},{\"post_id\":9154,\"post_title\":\"Recursive Functions\"}],\"course_link\":\"https:\/\/www.geeksforgeeks.org\/courses\/Data-Structures-With-Python\",\"views\":\"40\",\"video_schema\":{\"@context\":\"https:\/\/schema.org\",\"@type\":\"VideoObject\",\"name\":\"Introduction of Recursion\",\"description\":\"Introduction of Recursion in ProgrammingIn this video, we explore the concept of Recursion in Programming, a technique where a function calls itself in order to solve a problem. Recursion is a powerful and elegant approach to solving complex problems by breaking them down into smaller, more manageable sub-problems. In this tutorial, well cover the basics of recursion, its key principles, and how to implement it in your own programs.Key Features of RecursionSelf-Referencing A recursive function calls itself with modified parameters to work on smaller instances of the problem.Base Case Recursion relies on a base case to terminate the recursive calls and prevent infinite loops.Simplification of Problems Recursion is particularly effective for problems that can be naturally divided into smaller sub-problems, such as tree traversal, sorting algorithms, and more.Steps to Implement RecursionIdentify the Base Case Determine the simplest instance of the problem that can be solved directly without further recursion.Define the Recursive Case Define how the function should break the problem into smaller sub-problems and call itself with those smaller instances.Ensure Termination Ensure that each recursive call progresses toward the base case to avoid infinite recursion.Common Mistakes to AvoidMissing Base Case Failing to include a base case is a common error that can lead to infinite recursion and stack overflow.Excessive Recursion Depth Recursive solutions can sometimes exceed the maximum recursion depth, leading to a stack overflow. Be mindful of the problem size.Not Breaking Down the Problem Ensure that each recursive step makes progress toward solving a smaller instance of the problem.Applications of RecursionTree Traversal Recursion is widely used in traversing data structures like trees and graphs.Sorting Algorithms Algorithms like Merge Sort and Quick Sort use recursion to break down the problem into smaller sub-problems.Dynamic Programming Many dynamic programming problems are solved using recursion to build up solutions from smaller sub-problems.Backtracking Algorithms Problems like Sudoku, N-Queens, and other puzzle-solving algorithms are often solved using recursion.Why Learn Recursion?Understanding recursion is crucial for developing an in-depth understanding of algorithms and data structures. By mastering recursion, you willEnhance Problem-Solving Skills Learn how to solve complex problems by breaking them down into smaller, manageable sub-problems.Boost Algorithmic Thinking Recursion is a core concept used in many classic algorithms, and understanding it will significantly improve your ability to tackle algorithmic challenges.Deepen Your Programming Knowledge Recursion is widely used in various programming paradigms, and knowing it will allow you to write more efficient and elegant code.Topics CoveredBasics of Recursion Understand the fundamental concepts of recursion and its components.Recursive Functions Learn how to implement recursive functions in different programming languages.Real-World Examples Discover real-world applications of recursion in solving problems like tree traversal and sorting.\",\"thumbnailUrl\":[\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804.jpg\",\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804-small.png\",\"https:\/\/media.geeksforgeeks.org\/courses\/IntroductionofRecursion\/IntroductionofRecursion20241106130804-seo.png\"],\"uploadDate\":\"2024-11-06T13:08:36Z\",\"duration\":\"PT0H10M21S\",\"contentUrl\":\"https:\/\/www.geeksforgeeks.org\/videos\/introduction-of-recursion\/\"}}"); </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"VideoObject","name":"Introduction of Recursion","description":"Introduction of Recursion in ProgrammingIn this video, we explore the concept of Recursion in Programming, a technique where a function calls itself in order to solve a problem. Recursion is a powerful and elegant approach to solving complex problems by breaking them down into smaller, more manageable sub-problems. In this tutorial, well cover the basics of recursion, its key principles, and how to implement it in your own programs.Key Features of RecursionSelf-Referencing A recursive function calls itself with modified parameters to work on smaller instances of the problem.Base Case Recursion relies on a base case to terminate the recursive calls and prevent infinite loops.Simplification of Problems Recursion is particularly effective for problems that can be naturally divided into smaller sub-problems, such as tree traversal, sorting algorithms, and more.Steps to Implement RecursionIdentify the Base Case Determine the simplest instance of the problem that can be solved directly without further recursion.Define the Recursive Case Define how the function should break the problem into smaller sub-problems and call itself with those smaller instances.Ensure Termination Ensure that each recursive call progresses toward the base case to avoid infinite recursion.Common Mistakes to AvoidMissing Base Case Failing to include a base case is a common error that can lead to infinite recursion and stack overflow.Excessive Recursion Depth Recursive solutions can sometimes exceed the maximum recursion depth, leading to a stack overflow. Be mindful of the problem size.Not Breaking Down the Problem Ensure that each recursive step makes progress toward solving a smaller instance of the problem.Applications of RecursionTree Traversal Recursion is widely used in traversing data structures like trees and graphs.Sorting Algorithms Algorithms like Merge Sort and Quick Sort use recursion to break down the problem into smaller sub-problems.Dynamic Programming Many dynamic programming problems are solved using recursion to build up solutions from smaller sub-problems.Backtracking Algorithms Problems like Sudoku, N-Queens, and other puzzle-solving algorithms are often solved using recursion.Why Learn Recursion?Understanding recursion is crucial for developing an in-depth understanding of algorithms and data structures. 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class="three_dot_dropdown_div three_dot_dropdown_reportarticle"> <div class="three_dot_dropdown_inner_div" data-gfg-action="report-article" onclick="report_article();"> Report </div> </div>  </div> <div class="article--viewer_Gnews tooltip"> <a class="three_dot_dropdown_inner_div Gnews_wrapper" href ='https://news.google.com/publications/CAAqBwgKMLTrzwsw44bnAw?hl=en-IN&gl=IN&ceid=IN%3Aen' target="_blank"> <img class="gfg-icon_Gnews no-zoom-in-cursor" src=https://media.geeksforgeeks.org/auth-dashboard-uploads/Google-news.svg loading="lazy" alt="News"></img> Follow </a> </div> </div> </div> <div class="text"> What is Recursion?  The process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. Using a recursive algorithm, certain problems can be solved quite easily. Examples of such problems are <a href="https://www.geeksforgeeks.org/c-program-for-tower-of-hanoi/" rel="noopener">Towers of Hanoi (TOH)</a>, <a href="https://www.geeksforgeeks.org/tree-traversals-inorder-preorder-and-postorder/" rel="noopener">Inorder/Preorder/Postorder Tree Traversals</a>, <a href="https://www.geeksforgeeks.org/depth-first-traversal-for-a-graph/" rel="noopener">DFS of Graph</a>, etc. A recursive function solves a particular problem by calling itself with different parameters to solve smaller subproblems of the original problem. The called function may further call itself and this process might continue forever. So it is essential to know that we should provide a certain case in order to terminate this recursion process. We can say that every time the function calls itself with a simpler version of the original problem and moves toward the termination case or base case. Need of Recursion<div id="GFG_AD_gfg_mobile_336x280" style="margin: 5px 0;"></div> Recursion is an amazing technique with the help of which we can reduce the length of our code and make it easier to read and write. It has certain advantages over the iteration technique which will be discussed later. A task that can be defined with its similar subtask, recursion is one of the best solutions for it. For example; The Factorial of a number. Properties of Recursion: <ul><li value="1">Performing the same operations multiple times with different inputs.</li><li value="2">In every step, we try smaller inputs to make the problem smaller.</li><li value="3">Base condition is needed to stop the recursion otherwise infinite loop will occur.</li></ul><h4>Algorithm: Steps</h4><blockquote>Step1 – Define a base case: Identify the simplest case for which the solution is known or trivial. This is the stopping condition for the recursion, as it prevents the function from infinitely calling itself. Step2 – Define a recursive case: Define the problem in terms of smaller subproblems. Break the problem down into smaller versions of itself, and call the function recursively to solve each subproblem. Step3 – Ensure the recursion terminates: Make sure that the recursive function eventually reaches the base case, and does not enter an infinite loop. step4 – Combine the solutions: Combine the solutions of the subproblems to solve the original problem. </blockquote>A Mathematical Interpretation Let us consider a problem that a programmer has to determine the sum of first n natural numbers, there are several ways of doing that but the simplest approach is simply to add the numbers starting from 1 to n. So the function simply looks like this,<div id="GFG_AD_Desktop_InContent_ATF_336x280" style="text-align:center; max-height: 280px;"></div><div id="GFG_AD_gfg_mobile_336x280_1" style="margin: 5px 0;"></div> <blockquote>approach(1) – Simply adding one by one<div id="_GFG_ABP_Incontent_728x90" style="text-align:center;"></div> f(n) = 1 + 2 + 3 +……..+ n </blockquote>but there is another mathematical approach of representing this, <blockquote>approach(2) – Recursive adding  f(n) = 1                  n=1 f(n) = n + f(n-1)    n>1<div style="text-align:center;margin:20px 0; max-height: 280px;" id="GFG_AD_gfg_outstream_incontent"></div> </blockquote>There is a simple difference between the approach (1) and approach(2) and that is in approach(2) the function “ f( ) ” itself is being called inside the function, so this phenomenon is named recursion, and the function containing recursion is called recursive function, at the end, this is a great tool in the hand of the programmers to code some problems in a lot easier and efficient way. How are recursive functions stored in memory? Recursion uses more memory, because the recursive function adds to the stack with each recursive call, and keeps the values there until the call is finished. The recursive function uses LIFO (LAST IN FIRST OUT) Structure just like the stack data structure. <a href="https://www.geeksforgeeks.org/stack-data-structure/" rel="noopener">https://www.geeksforgeeks.org/stack-data-structure/</a>   What is the base condition in recursion?  In the recursive program, the solution to the base case is provided and the solution to the bigger problem is expressed in terms of smaller problems.    <gfg-tabs data-run-ide="true" data-mode="light"> <gfg-tab slot="tab">C++</gfg-tab> <gfg-panel slot="panel" data-code-lang="cpp"> <code class="language-cpp"><div class=highlight><pre>#include <iostream> using namespace std; int fact(int n) { // BASE CONDITION if (n == 0 || n == 1) return 1; return n * fact(n - 1); } int main() { cout << "Factorial of 5 : " << fact(5); return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C</gfg-tab> <gfg-panel slot="panel" data-code-lang="c"> <code class="language-c"><div class=highlight><pre>#include <stdio.h> int fact(int n) { // BASE CONDITION if (n == 0 || n == 1) return 1; return n * fact(n - 1); } int main() { printf("Factorial of 5 : %d\n", fact(5)); return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Java</gfg-tab> <gfg-panel slot="panel" data-code-lang="java"> <code class="language-java"><div class=highlight><pre>public class GfG { public static int fact(int n) { // BASE CONDITION if (n == 0 || n == 1) return 1; return n * fact(n - 1); } public static void main(String[] args) { System.out.println("Factorial of 5 : " + fact(5)); } } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Python</gfg-tab> <gfg-panel slot="panel" data-code-lang="python3"> <code class="language-python3"><div class=highlight><pre>def fact(n): # BASE CONDITION if n == 0 or n == 1: return 1 return n * fact(n - 1) print("Factorial of 5 : ", fact(5)) </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C#</gfg-tab> <gfg-panel slot="panel" data-code-lang="csharp"> <code class="language-csharp"><div class=highlight><pre>using System; class Program { static int Fact(int n) { // BASE CONDITION if (n == 0 || n == 1) return 1; return n * Fact(n - 1); } static void Main() { Console.WriteLine("Factorial of 5 : " + Fact(5)); } } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">JavaScript</gfg-tab> <gfg-panel slot="panel" data-code-lang="javascript"> <code class="language-javascript"><div class=highlight><pre>function fact(n) { // BASE CONDITION if (n === 0 || n === 1) return 1; return n * fact(n - 1); } console.log("Factorial of 5 : " + fact(5)); </pre></div></code> </gfg-panel> <gfg-tab slot="tab">PHP</gfg-tab> <gfg-panel slot="panel" data-code-lang="php"> <code class="language-php"><div class=highlight><pre><?php function fact($n) { // BASE CONDITION if ($n == 0 || $n == 1) return 1; return $n * fact($n - 1); } echo "Factorial of 5 : " . fact(5); ?> </pre></div></code> </gfg-panel></gfg-tabs> In the above example, the base case for n < = 1 is defined and the larger value of a number can be solved by converting to a smaller one till the base case is reached. How a particular problem is solved using recursion?  The idea is to represent a problem in terms of one or more smaller problems, and add one or more base conditions that stop the recursion. For example, we compute factorial n if we know the factorial of (n-1). The base case for factorial would be n = 0. We return 1 when n = 0.  Why Stack Overflow error occurs in recursion?  If the base case is not reached or not defined, then the stack overflow problem may arise. Let us take an example to understand this. <pre>int fact(int n) { // wrong base case (it may cause // stack overflow). if (n == 100) return 1; else return n*fact(n-1); }</pre>If fact(10) is called, it will call fact(9), fact(8), fact(7), and so on but the number will never reach 100. So, the base case is not reached. If the memory is exhausted by these functions on the stack, it will cause a stack overflow error.  What is the difference between direct and indirect recursion?  A function fun is called direct recursive if it calls the same function fun. A function fun is called indirect recursive if it calls another function say fun_new and fun_new calls fun directly or indirectly. The difference between direct and indirect recursion has been illustrated in Table 1. <div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-1"></div> <pre>// An example of direct recursion void directRecFun() { // Some code.... directRecFun(); // Some code... } // An example of indirect recursion void indirectRecFun1() { // Some code... indirectRecFun2(); // Some code... } void indirectRecFun2() { // Some code... indirectRecFun1(); // Some code... }</pre>What is the difference between tailed and non-tailed recursion?  A recursive function is tail recursive when a recursive call is the last thing executed by the function. Please refer <a href="https://www.geeksforgeeks.org/tail-recursion/" rel="noopener">tail recursion article</a> for details.  How memory is allocated to different function calls in recursion?  When any function is called from main(), the memory is allocated to it on the stack. A recursive function calls itself, the memory for a called function is allocated on top of memory allocated to the calling function and a different copy of local variables is created for each function call. When the base case is reached, the function returns its value to the function by whom it is called and memory is de-allocated and the process continues. Let us take the example of how recursion works by taking a simple function.  <gfg-tabs data-run-ide="true" data-mode="light"> <gfg-tab slot="tab">C++</gfg-tab> <gfg-panel slot="panel" data-code-lang="cpp"> <code class="language-cpp"><div class=highlight><pre>// A C++ program to demonstrate working of // recursion #include <bits/stdc++.h> using namespace std; void printFun(int test) { if (test < 1) return; else { cout << test << " "; printFun(test - 1); // statement 2 cout << test << " "; return; } } // Driver Code int main() { int test = 3; printFun(test); } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C</gfg-tab> <gfg-panel slot="panel" data-code-lang="c"> <code class="language-c"><div class=highlight><pre>// A C program to demonstrate working of recursion #include <stdio.h> void printFun(int test) { if (test < 1) return; else { printf("%d ", test); printFun(test - 1); // statement 2 printf("%d ", test); return; } } // Driver Code int main() { int test = 3; printFun(test); return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Java</gfg-tab> <gfg-panel slot="panel" data-code-lang="java"> <code class="language-java"><div class=highlight><pre>// A Java program to demonstrate working of // recursion class GFG { static void printFun(int test) { if (test < 1) return; else { System.out.printf("%d ", test); printFun(test - 1); // statement 2 System.out.printf("%d ", test); return; } } // Driver Code public static void main(String[] args) { int test = 3; printFun(test); } } // This code is contributed by // Smitha Dinesh Semwal </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Python</gfg-tab> <gfg-panel slot="panel" data-code-lang="python3"> <code class="language-python3"><div class=highlight><pre># A Python 3 program to # demonstrate working of # recursion def printFun(test): if (test < 1): return else: print(test, end=" ") printFun(test-1) # statement 2 print(test, end=" ") return # Driver Code test = 3 printFun(test) # This code is contributed by # Smitha Dinesh Semwal </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C#</gfg-tab> <gfg-panel slot="panel" data-code-lang="csharp"> <code class="language-csharp"><div class=highlight><pre>// A C# program to demonstrate // working of recursion using System; class GFG { // function to demonstrate // working of recursion static void printFun(int test) { if (test < 1) return; else { Console.Write(test + " "); // statement 2 printFun(test - 1); Console.Write(test + " "); return; } } // Driver Code public static void Main(String[] args) { int test = 3; printFun(test); } } // This code is contributed by Anshul Aggarwal. </pre></div></code> </gfg-panel> <gfg-tab slot="tab">JavaScript</gfg-tab> <gfg-panel slot="panel" data-code-lang="javascript"> <code class="language-javascript"><div class=highlight><pre>// A JavaScript program to demonstrate working of recursion function printFun(test) { if (test < 1) return; else { console.log(test); printFun(test - 1); // statement 2 console.log(test); return; } } // Driver Code let test = 3; printFun(test); </pre></div></code> </gfg-panel> <gfg-tab slot="tab">PHP</gfg-tab> <gfg-panel slot="panel" data-code-lang="php"> <code class="language-php"><div class=highlight><pre><?php // PHP program to demonstrate // working of recursion // function to demonstrate // working of recursion function printFun($test) { if ($test < 1) return; else { echo("$test "); // statement 2 printFun($test-1); echo("$test "); return; } } // Driver Code $test = 3; printFun($test); // This code is contributed by // Smitha Dinesh Semwal. ?> </pre></div></code> </gfg-panel></gfg-tabs><div class="code-output"> Output<pre>3 2 1 1 2 3 </pre></div>Time Complexity: O(1) Auxiliary Space: O(1) When printFun(3) is called from main(), memory is allocated to printFun(3) and a local variable test is initialized to 3 and statement 1 to 4 are pushed on the stack as shown in below diagram. It first prints ‘3’. In statement 2, printFun(2) is called and memory is allocated to printFun(2) and a local variable test is initialized to 2 and statement 1 to 4 are pushed into the stack. Similarly, printFun(2) calls printFun(1) and printFun(1) calls printFun(0). printFun(0) goes to if statement and it return to printFun(1). The remaining statements of printFun(1) are executed and it returns to printFun(2) and so on. In the output, values from 3 to 1 are printed and then 1 to 3 are printed. The memory stack has been shown in below diagram. <img src="https://media.geeksforgeeks.org/wp-content/cdn-uploads/recursion.jpg" alt="recursion" width="inherit" height="inherit"> Recursion VS Iteration <table><thead></thead><tbody><tr><td>SR No.</td><td>Recursion</td><td>Iteration</td></tr><tr><td>1)</td><td>Terminates when the base case becomes true.</td><td>Terminates when the condition becomes false.</td></tr><tr><td>2)</td><td>Used with functions.</td><td>Used with loops.</td></tr><tr><td>3)</td><td>Every recursive call needs extra space in the stack memory.</td><td>Every iteration does not require any extra space.</td></tr><tr><td>4)</td><td>Smaller code size.</td><td>Larger code size.</td></tr></tbody></table>Now, let’s discuss a few practical problems which can be solved by using recursion and understand its basic working. For basic understanding please read the following articles.  <a href="https://www.geeksforgeeks.org/recursion/" rel="noopener">Basic understanding of Recursion.</a> Problem 1: Write a program and recurrence relation to find the Fibonacci series of n where n>2 .  Mathematical Equation:   <pre>n if n == 0, n == 1; fib(n) = fib(n-1) + fib(n-2) otherwise;</pre>Recurrence Relation:  <pre>T(n) = T(n-1) + T(n-2) + O(1)</pre>Recursive program:  <pre>Input: n = 5 Output: Fibonacci series of 5 numbers is : 0 1 1 2 3</pre><gfg-tabs data-run-ide="true" data-mode="light"> <gfg-tab slot="tab">C++</gfg-tab> <gfg-panel slot="panel" data-code-lang="cpp"> <code class="language-cpp"><div class=highlight><pre>// C++ code to implement Fibonacci series #include <bits/stdc++.h> using namespace std; // Function for fibonacci int fib(int n) { // Stop condition if (n == 0) return 0; // Stop condition if (n == 1 || n == 2) return 1; // Recursion function else return (fib(n - 1) + fib(n - 2)); } // Driver Code int main() { // Initialize variable n. int n = 5; cout<<"Fibonacci series of 5 numbers is: "; // for loop to print the fibonacci series. for (int i = 0; i < n; i++) { cout<<fib(i)<<" "; } return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C</gfg-tab> <gfg-panel slot="panel" data-code-lang="c"> <code class="language-c"><div class=highlight><pre>// C code to implement Fibonacci series #include <stdio.h> // Function for fibonacci int fib(int n) { // Stop condition if (n == 0) return 0; // Stop condition if (n == 1 || n == 2) return 1; // Recursion function else return (fib(n - 1) + fib(n - 2)); } // Driver Code int main() { // Initialize variable n. int n = 5; printf("Fibonacci series " "of %d numbers is: ", n); // for loop to print the fibonacci series. for (int i = 0; i < n; i++) { printf("%d ", fib(i)); } return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Java</gfg-tab> <gfg-panel slot="panel" data-code-lang="java"> <code class="language-java"><div class=highlight><pre>// Java code to implement Fibonacci series import java.util.*; class GFG { // Function for fibonacci static int fib(int n) { // Stop condition if (n == 0) return 0; // Stop condition if (n == 1 || n == 2) return 1; // Recursion function else return (fib(n - 1) + fib(n - 2)); } // Driver Code public static void main(String []args) { // Initialize variable n. int n = 5; System.out.print("Fibonacci series of 5 numbers is: "); // for loop to print the fibonacci series. for (int i = 0; i < n; i++) { System.out.print(fib(i)+" "); } } } // This code is contributed by rutvik_56. </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Python</gfg-tab> <gfg-panel slot="panel" data-code-lang="python3"> <code class="language-python3"><div class=highlight><pre># Python code to implement Fibonacci series # Function for fibonacci def fib(n): # Stop condition if (n == 0): return 0 # Stop condition if (n == 1 or n == 2): return 1 # Recursion function else: return (fib(n - 1) + fib(n - 2)) # Driver Code # Initialize variable n. n = 5; print("Fibonacci series of 5 numbers is :",end=" ") # for loop to print the fibonacci series. for i in range(0,n): print(fib(i),end=" ") </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C#</gfg-tab> <gfg-panel slot="panel" data-code-lang="csharp"> <code class="language-csharp"><div class=highlight><pre>using System; public class GFG { // Function for fibonacci static int fib(int n) { // Stop condition if (n == 0) return 0; // Stop condition if (n == 1 || n == 2) return 1; // Recursion function else return (fib(n - 1) + fib(n - 2)); } // Driver Code static public void Main () { // Initialize variable n. int n = 5; Console.Write("Fibonacci series of 5 numbers is: "); // for loop to print the fibonacci series. for (int i = 0; i < n; i++) { Console.Write(fib(i) + " "); } } } // This code is contributed by avanitrachhadiya2155 </pre></div></code> </gfg-panel> <gfg-tab slot="tab">JavaScript</gfg-tab> <gfg-panel slot="panel" data-code-lang="javascript"> <code class="language-javascript"><div class=highlight><pre>// Function for fibonacci function fib(n) { // Stop condition if (n === 0) return 0; // Stop condition if (n === 1 || n === 2) return 1; // Recursion function return fib(n - 1) + fib(n - 2); } // Driver Code let n = 5; console.log("Fibonacci series of 5 numbers is:"); // for loop to print the fibonacci series. for (let i = 0; i < n; i++) { console.log(fib(i) + " "); } </pre></div></code> </gfg-panel></gfg-tabs><div class="code-output"> Output<pre>Fibonacci series of 5 numbers is: 0 1 1 2 3 </pre></div>Time Complexity: O(2n) Auxiliary Space: O(n)<div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-2"></div> Here is the recursive tree for input 5 which shows a clear picture of how a big problem can be solved into smaller ones.  fib(n) is a Fibonacci function. The time complexity of the given program can depend on the function call.  <blockquote>fib(n) -> level CBT (UB) -> 2^n-1 nodes -> 2^n function call -> 2^n*O(1) -> T(n) = O(2^n)   </blockquote>For Best Case.  <pre>T(n) = θ(2^n\2)</pre>Working:  <img src="https://media.geeksforgeeks.org/wp-content/uploads/20191107235734/fib1.jpg" width="inherit" height="inherit" loading="lazy"> Problem 2: Write a program and recurrence relation to find the Factorial of n where n>2 .  Mathematical Equation:  <pre>1 if n == 0 or n == 1; f(n) = n*f(n-1) if n> 1;</pre>Recurrence Relation:  <pre>T(n) = 1 for n = 0 T(n) = 1 + T(n-1) for n > 0</pre>Recursive Program:  Input: n = 5  Output:  factorial of 5 is: 120 Implementation:  <gfg-tabs data-run-ide="true" data-mode="light"> <gfg-tab slot="tab">C++</gfg-tab> <gfg-panel slot="panel" data-code-lang="cpp"> <code class="language-cpp"><div class=highlight><pre>// C++ code to implement factorial #include <bits/stdc++.h> using namespace std; // Factorial function int f(int n) { // Stop condition if (n == 0 || n == 1) return 1; // Recursive condition else return n * f(n - 1); } // Driver code int main() { int n = 5; cout<<"factorial of "<<n<<" is: "<<f(n); return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C</gfg-tab> <gfg-panel slot="panel" data-code-lang="c"> <code class="language-c"><div class=highlight><pre>// C code to implement factorial #include <stdio.h> // Factorial function int f(int n) { // Stop condition if (n == 0 || n == 1) return 1; // Recursive condition else return n * f(n - 1); } // Driver code int main() { int n = 5; printf("factorial of %d is: %d", n, f(n)); return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Java</gfg-tab> <gfg-panel slot="panel" data-code-lang="java"> <code class="language-java"><div class=highlight><pre>// Java code to implement factorial public class GFG { // Factorial function static int f(int n) { // Stop condition if (n == 0 || n == 1) return 1; // Recursive condition else return n * f(n - 1); } // Driver code public static void main(String[] args) { int n = 5; System.out.println("factorial of " + n + " is: " + f(n)); } } // This code is contributed by divyesh072019. </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Python</gfg-tab> <gfg-panel slot="panel" data-code-lang="python3"> <code class="language-python3"><div class=highlight><pre># Python3 code to implement factorial # Factorial function def f(n): # Stop condition if (n == 0 or n == 1): return 1; # Recursive condition else: return n * f(n - 1); # Driver code if __name__=='__main__': n = 5; print("factorial of",n,"is:",f(n)) # This code is contributed by pratham76. </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C#</gfg-tab> <gfg-panel slot="panel" data-code-lang="csharp"> <code class="language-csharp"><div class=highlight><pre>// C# code to implement factorial using System; class GFG { // Factorial function static int f(int n) { // Stop condition if (n == 0 || n == 1) return 1; // Recursive condition else return n * f(n - 1); } // Driver code static void Main() { int n = 5; Console.WriteLine("factorial of " + n + " is: " + f(n)); } } // This code is contributed by divyeshrabadiya07. </pre></div></code> </gfg-panel> <gfg-tab slot="tab">JavaScript</gfg-tab> <gfg-panel slot="panel" data-code-lang="javascript"> <code class="language-javascript"><div class=highlight><pre>// Factorial function function f(n) { // Stop condition if (n === 0 || n === 1) return 1; // Recursive condition else return n * f(n - 1); } // Driver code let n = 5; console.log(`factorial of ${n} is: ${f(n)}`); </pre></div></code> </gfg-panel></gfg-tabs><div class="code-output"> Output<pre>factorial of 5 is: 120</pre></div>Time complexity: O(n) Auxiliary Space: O(n) Working:   <div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-3"></div> <div style="width: 810px" class="wp-caption alignnone"><a href="https://media.geeksforgeeks.org/wp-content/uploads/20191108122114/Untitled-Diagram58.jpg" target="_blank" rel="noopener noreferrer nofollow"><img src="https://media.geeksforgeeks.org/wp-content/uploads/20191108122114/Untitled-Diagram58.jpg" width="inherit" height="inherit" loading="lazy"></a>Diagram of factorial Recursion function for user input 5.</div><h3>Example:  Real Applications of Recursion in real problems</h3>Recursion is a powerful technique that has many applications in computer science and programming. Here are some of the common applications of recursion: <ul><li value="1">Tree and graph traversal: Recursion is frequently used for traversing and searching data structures such as trees and graphs. Recursive algorithms can be used to explore all the nodes or vertices of a tree or graph in a systematic way.</li><li value="2">Sorting algorithms: Recursive algorithms are also used in sorting algorithms such as quicksort and merge sort. These algorithms use recursion to divide the data into smaller subarrays or sublists, sort them, and then merge them back together.</li><li value="3">Divide-and-conquer algorithms: Many algorithms that use a divide-and-conquer approach, such as the binary search algorithm, use recursion to break down the problem into smaller subproblems.</li><li value="4">Fractal generation: Fractal shapes and patterns can be generated using recursive algorithms. For example, the Mandelbrot set is generated by repeatedly applying a recursive formula to complex numbers.</li><li value="5">Backtracking algorithms: Backtracking algorithms are used to solve problems that involve making a sequence of decisions, where each decision depends on the previous ones. These algorithms can be implemented using recursion to explore all possible paths and backtrack when a solution is not found.</li><li value="6">Memoization: Memoization is a technique that involves storing the results of expensive function calls and returning the cached result when the same inputs occur again. Memoization can be implemented using recursive functions to compute and cache the results of subproblems.</li></ul>These are just a few examples of the many applications of recursion in computer science and programming. Recursion is a versatile and powerful tool that can be used to solve many different types of problems. <h3>Explanation: one real example of recursion:</h3>Recursion is a programming technique that involves a function calling itself. It can be a powerful tool for solving complex problems, but it also requires careful implementation to avoid infinite loops and stack overflows.  <gfg-tabs data-run-ide="true" data-mode="light"> <gfg-tab slot="tab">C++</gfg-tab> <gfg-panel slot="panel" data-code-lang="cpp"> <code class="language-cpp"><div class=highlight><pre>#include <iostream> using namespace std; int factorial(int n) { // Base case: if n is 0 or 1, return 1 if (n == 0 || n == 1) { return 1; } // Recursive case: if n is greater than 1, // call the function with n-1 and multiply by n else { return n * factorial(n - 1); } } int main() { // Call the factorial function and print the result int result = factorial(5); cout << result <<endl; // Output: 120 return 0; } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Java</gfg-tab> <gfg-panel slot="panel" data-code-lang="java"> <code class="language-java"><div class=highlight><pre>import java.util.*; class Main { public static int factorial(int n) { // Base case: if n is 0 or 1, return 1 if (n == 0 || n == 1) { return 1; } // Recursive case: if n is greater than 1, // call the function with n-1 and multiply by n else { return n * factorial(n - 1); } } public static void main(String[] args) { // Call the factorial function and print the result int result = factorial(5); System.out.println(result); // Output: 120 } } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">Python</gfg-tab> <gfg-panel slot="panel" data-code-lang="python3"> <code class="language-python3"><div class=highlight><pre>def factorial(n): # Base case: if n is 0 or 1, return 1 if n == 0 or n == 1: return 1 # Recursive case: if n is greater than 1, call the function with n-1 and multiply by n else: return n * factorial(n-1) # Call the factorial function and print the result result = factorial(5) print(result) # Output: 120 </pre></div></code> </gfg-panel> <gfg-tab slot="tab">C#</gfg-tab> <gfg-panel slot="panel" data-code-lang="csharp"> <code class="language-csharp"><div class=highlight><pre>using System; class MainClass { public static int factorial(int n) { // Base case: if n is 0 or 1, return 1 if (n == 0 || n == 1) { return 1; } // Recursive case: if n is greater than 1, // call the function with n-1 and multiply by n else { return n * factorial(n - 1); } } public static void Main (string[] args) { // Call the factorial function and print the result int result = factorial(5); Console.WriteLine(result); // Output: 120 } } </pre></div></code> </gfg-panel> <gfg-tab slot="tab">JavaScript</gfg-tab> <gfg-panel slot="panel" data-code-lang="javascript"> <code class="language-javascript"><div class=highlight><pre>function factorial(n) { // Base case: if n is 0 or 1, return 1 if (n === 0 || n === 1) { return 1; } // Recursive case: if n is greater than 1, call the function with n-1 and multiply by n else { return n * factorial(n - 1); } } // Call the factorial function and print the result const result = factorial(5); console.log(result); // Output: 120 </pre></div></code> </gfg-panel></gfg-tabs><div class="code-output"> Output<pre>120</pre></div>In this example, we define a function called factorial that takes an integer n as input. The function uses recursion to compute the factorial of n (i.e., the product of all positive integers up to n). The factorial function first checks if n is 0 or 1, which are the base cases. If n is 0 or 1, the function returns 1, since 0! and 1! are both 1. If n is greater than 1, the function enters the recursive case. It calls itself with n-1 as the argument and multiplies the result by n. This computes n! by recursively computing (n-1)!. It’s important to note that recursion can be inefficient and lead to stack overflows if not used carefully. Each function call adds a new frame to the call stack, which can cause the stack to grow too large if the recursion is too deep. In addition, recursion can make the code more difficult to understand and debug, since it requires thinking about multiple levels of function calls. However, recursion can also be a powerful tool for solving complex problems, particularly those that involve breaking a problem down into smaller subproblems. When used correctly, recursion can make the code more elegant and easier to read. What are the disadvantages of recursive programming over iterative programming?  Note that both recursive and iterative programs have the same problem-solving powers, i.e., every recursive program can be written iteratively and vice versa is also true. The recursive program has greater space requirements than the iterative program as all functions will remain in the stack until the base case is reached. It also has greater time requirements because of function calls and returns overhead. Moreover, due to the smaller length of code, the codes are difficult to understand and hence extra care has to be practiced while writing the code. The computer may run out of memory if the recursive calls are not properly checked. What are the advantages of recursive programming over iterative programming?  Recursion provides a clean and simple way to write code. Some problems are inherently recursive like tree traversals, <a href="https://www.geeksforgeeks.org/c-program-for-tower-of-hanoi/" rel="noopener">Tower of Hanoi</a>, etc. For such problems, it is preferred to write recursive code. We can write such codes also iteratively with the help of a stack data structure. For example refer <a href="https://www.geeksforgeeks.org/inorder-tree-traversal-without-recursion/" rel="noopener">Inorder Tree Traversal without Recursion</a>, <a href="https://www.geeksforgeeks.org/iterative-tower-of-hanoi/" rel="noopener">Iterative Tower of Hanoi</a>. Summary of Recursion: <ul><li value="1">There are two types of cases in recursion i.e. recursive case and a base case.</li><li value="2">The base case is used to terminate the recursive function when the case turns out to be true.</li><li value="3">Each recursive call makes a new copy of that method in the stack memory.</li><li value="4">Infinite recursion may lead to running out of stack memory.</li><li value="5">Examples of Recursive algorithms: Merge Sort, Quick Sort, Tower of Hanoi, Fibonacci Series, Factorial Problem, etc.</li></ul>Output based practice problems for beginners:  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion/" rel="noopener">Practice Questions for Recursion | Set 1</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-2/" rel="noopener">Practice Questions for Recursion | Set 2</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-3/" rel="noopener">Practice Questions for Recursion | Set 3</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-4/" rel="noopener">Practice Questions for Recursion | Set 4</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-5/" rel="noopener">Practice Questions for Recursion | Set 5</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-6/" rel="noopener">Practice Questions for Recursion | Set 6</a>  <a href="https://www.geeksforgeeks.org/practice-questions-for-recursion-set-7/" rel="noopener">Practice Questions for Recursion | Set 7</a> <a href="https://www.geeksforgeeks.org/algorithms-gq/recursion-gq/" rel="noopener">Quiz on Recursion</a>  <a href="https://www.geeksforgeeks.org/explore?page=1&category=Recursion&sortBy=submissions" rel="noopener">Coding Practice on Recursion:</a>  <a href="https://www.geeksforgeeks.org/tag/recursion/" rel="noopener">All Articles on Recursion</a>  <a href="https://www.geeksforgeeks.org/recursion-practice-problems-solutions/" rel="noopener">Recursive Practice Problems with Solutions</a> <div id="AP_G4GR_6"></div> <div class='article_bottom_text'></div> <div id="video-tab-content" class="video-tab-content"> <div style="text-align: center; margin: 20px 0px;" id="GFG_AD_InContent_Desktop_728x280"></div> </div> <div class="article-meta-author-details"> <div class="article-meta-author-details-block"> <div class="article-meta-author-details-profile-display"> <div class="article-meta-author-details-profile-display-icon"> <div class="image-wrap auth-external-author" style="position: relative; pointer-events: none; user-select: none;margin-top: 6px;">S</div> </div> <div class="article-meta-author-details-profile-display-name" > <div class="info" style="position: unset;"> <div class="name" style="cursor: default;"> Sonal Tuteja </div> </div> </div> </div> </div> <div class="article_bottom_suggestion_wrapper"> <div class="article_bottom_suggestion" data-title="Follow"> <a href='https://news.google.com/publications/CAAqBwgKMLTrzwsw44bnAw?hl=en-IN&gl=IN&ceid=IN%3Aen' target="_blank"> <img class="gfg-icon_Gnews no-zoom-in-cursor" src=https://media.geeksforgeeks.org/auth-dashboard-uploads/Google-news.svg loading="lazy" alt="News"></img> </a> </div> <div class='discussion_panel' data-title='Comments'> </div> <div pid="140498" class="article--viewer_like tooltip tooltipBottom" data-title="Like Article">  <button id="likeButton" aria-label="like article" data-gfg-action="like-article" data-bookmark-value="0" data-liked="false" style="color: #5B5A5A !important; display: flex; align-items: center; background-color: unset; margin: 0px !important; padding: 5px 0px;"> <figure id="likeCount" style="margin-left: 3px; margin-top: 4px; color: var(--like-count-color); font-size: 14px; font-weight: 600;" class="favoriteText"></figure> </button> </div> <div class="article_bottom_suggestion" onclick="improveArticleCall()" data-title="Improve"> Improve </div> </div> </div> <div class="d-row content-bw "style="margin-top: 20px;"> <div class="article-pgnavi_prev"> </div> <div class="article-pgnavi_next"> <a href="https://www.geeksforgeeks.org/what-is-recursion/?ref=next_article" class="pg-head"> Next Article </a>  <div class="pg-main"> <a href="https://www.geeksforgeeks.org/what-is-recursion/?ref=next_article">What is Recursion?</a> </div> </div> </div> </div> </article> <div class="article--recommended article--recommended_wrapper" id="similar-reads"> <h3 class="new-top-bar top-bar-title">Similar Reads</h3> <div class="gfg-similar-reads-list"> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/why-is-tail-recursion-optimization-faster-than-normal-recursion/?ref=oin_asr1"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Why is Tail Recursion optimization faster than normal Recursion?</div> <div class="gfg-similar-read-item-subheading">What is tail recursion? 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Else reverse(head->p</div> </div> <div class="reading-time"> 9 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/sum-of-natural-numbers-using-recursion/?ref=oin_asr14"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Sum of natural numbers using recursion</div> <div class="gfg-similar-read-item-subheading">Given a number n, find sum of first n natural numbers. 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Examples: Input : N = 5 , M = 3 Output : 15 Input : N = 5 , M = -3 Output : -15 Input : N = -5 , M = 3 Output : -15 Input : N = -5 , M = -3 Output:15 A recursive solution to the above problem for only posit</div> </div> <div class="reading-time"> 8 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/program-to-check-if-an-array-is-palindrome-or-not-using-recursion/?ref=oin_asr23"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Program to check if an array is palindrome or not using Recursion</div> <div class="gfg-similar-read-item-subheading">Given an array. The task is to determine whether an array is a palindrome or not using recursion. Examples: Input: arr[] = {3, 6, 0, 6, 3}Output: Palindrome Input: arr[] = {1, 2, 3, 4, 5}Output: Not Palindrome Approach: Base case: If array has only one element i.e. begin == end then return 1, also if begin>end which means the array is palindrome</div> </div> <div class="reading-time"> 4 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/difference-between-recursion-and-iteration/?ref=oin_asr24"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Difference between Recursion and Iteration</div> <div class="gfg-similar-read-item-subheading">A program is called recursive when an entity calls itself. A program is called iterative when there is a loop (or repetition). Example: Program to find the factorial of a number C/C++ Code // C program to find factorial of given number #include <stdio.h> // ----- Recursion ----- // method to find factorial of given number int factorialUsingRe</div> </div> <div class="reading-time"> 8 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/delete-nodes-which-have-a-greater-value-on-right-side-using-recursion/?ref=oin_asr25"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Delete nodes which have a greater value on right side using recursion</div> <div class="gfg-similar-read-item-subheading">Given a singly linked list, remove all the nodes which have a greater value on the right side. Examples: a) The list 12->15->10->11->5->6->2->3->NULL should be changed to 15->11->6->3->NULL. Note that 12, 10, 5 and 2 have been deleted because there is a greater value on the right side. When we examine 12, we see</div> </div> <div class="reading-time"> 7 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/sum-of-the-series-11-22-33-nn-using-recursion/?ref=oin_asr26"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Sum of the series 1^1 + 2^2 + 3^3 + ..... + n^n using recursion</div> <div class="gfg-similar-read-item-subheading">Given an integer n, the task is to find the sum of the series 11 + 22 + 33 + ..... + nn using recursion. Examples: Input: n = 2 Output: 5 11 + 22 = 1 + 4 = 5 Input: n = 3 Output: 32 11 + 22 + 33 = 1 + 4 + 27 = 32 Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recur</div> </div> <div class="reading-time"> 3 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/find-hcf-of-two-numbers-without-using-recursion-or-euclidean-algorithm/?ref=oin_asr27"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Find HCF of two numbers without using recursion or Euclidean algorithm</div> <div class="gfg-similar-read-item-subheading">Given two integer x and y, the task is to find the HCF of the numbers without using recursion or Euclidean method. Examples: Input: x = 16, y = 32 Output: 16 Input: x = 12, y = 15 Output: 3 Approach: HCF of two numbers is the greatest number which can divide both the numbers. If the smaller of the two numbers can divide the larger number then the H</div> </div> <div class="reading-time"> 6 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/print-fibonacci-series-in-reverse-order-using-recursion/?ref=oin_asr28"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Print Fibonacci Series in reverse order using Recursion</div> <div class="gfg-similar-read-item-subheading">Given an integer N, the task is to print the first N terms of the Fibonacci series in reverse order using Recursion. Examples: Input: N = 5Output: 3 2 1 1 0Explanation: First five terms are - 0 1 1 2 3. Input: N = 10Output: 34 21 13 8 5 3 2 1 1 0 Approach: The idea is to use recursion in a way that keeps calling the same function again till N is gr</div> </div> <div class="reading-time"> 3 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/find-the-node-with-maximum-value-in-a-binary-search-tree-using-recursion/?ref=oin_asr29"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Find the node with maximum value in a Binary Search Tree using recursion</div> <div class="gfg-similar-read-item-subheading">Given a Binary Search Tree, the task is to find the node with maximum value. Examples: Input: Output: 22 Approach: Just traverse the node from root to right recursively until right is NULL. The node whose right is NULL is the node with the maximum value. Below is the implementation of the above approach: C/C++ Code // C++ implementation of the appr</div> </div> <div class="reading-time"> 7 min read </div> </a> <a class="gfg-similar-read-item" href="https://www.geeksforgeeks.org/programs-to-print-triangle-and-diamond-patterns-using-recursion/?ref=oin_asr30"> <div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Programs to print Triangle and Diamond patterns using recursion</div> <div class="gfg-similar-read-item-subheading">This article is aimed at giving a recursive implementation for pattern printing. Pattern 1: Example:Input: 5 Output: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *Implementation: C/C++ Code // Program to print the given pattern #include <iostream> using namespace std; void print_asterisk(int asterisk) { if (asterisk == 0) return;</div> </div> <div class="reading-time"> 15+ min read </div> </a> </div> </div> <div class="bottom-wrap" style="padding: 0px 20px;margin-bottom: 10px;"> <div class="improved"> <div class="t-head">Article Tags : </div> <ul> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/dsa/algorithm/?ref=article_category">Algorithms</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/dsa/?ref=article_category">DSA</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/category/dsa/algorithm/recursion/?ref=article_category">Recursion</a> </li> <li style="border-radius: 25px;" class=""> <a href="https://www.geeksforgeeks.org/tag/dsa-tutorials/?ref=article_tag">DSA Tutorials</a> </li> <li style="border-radius: 25px;display:none;" class="articles-hidden-tags"> <a href="https://www.geeksforgeeks.org/tag/tail-recursion/?ref=article_tag">tail-recursion</a> </li> </ul> <button id="showMoreTagsBtn" class="show-more-recommendation-button">+1 More</button> </div> <div class="improved"> <div class="t-head">Practice Tags : </div> <ul class="practice-tags"> <li><a href="https://www.geeksforgeeks.org/explore?category=Algorithms&ref=article_practice_tag">Algorithms</a></li><li><a href="https://www.geeksforgeeks.org/explore?category=Recursion&ref=article_practice_tag">Recursion</a></li> </ul> </div> </div> <div class="vote-wrap"> <div style="display:none;align-items:center;justify-content:center;width:100%;"> <button aria-label="like" data-type="like" class="vote-this" style="margin-right: 0; 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