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 recursive algorithm, certain problems can be solved quite easily. Examples of such problems are Towers of Hanoi (TOH), Inorder/Preorder/Postorder Tree Traversals, DFS of Graph, etc.
Types of Recursions:
Recursion are mainly of two types depending on whether a function calls itself from within itself or more than one function call one another mutually. The first one is called direct recursion and another one is called indirect recursion. Thus, the two types of recursion are:
1. Direct Recursion: These can be further categorized into four types:
- Tail Recursion: If a recursive function calling itself and that recursive call is the last statement in the function then it's known as Tail Recursion. After that call the recursive function performs nothing. The function has to process or perform any operation at the time of calling and it does nothing at returning time.
Example:
C++
// Code Showing Tail Recursion
#include <iostream>
using namespace std;
// Recursion function
void fun(int n)
{
if (n > 0) {
cout << n << " ";
// Last statement in the function
fun(n - 1);
}
}
// Driver Code
int main()
{
int x = 3;
fun(x);
return 0;
}
// This code is contributed by shubhamsingh10
// Code Showing Tail Recursion
#include <stdio.h>
// Recursion function
void fun(int n)
{
if (n > 0) {
printf("%d ", n);
// Last statement in the function
fun(n - 1);
}
}
// Driver Code
int main()
{
int x = 3;
fun(x);
return 0;
}
// Java code Showing Tail Recursion
class GFG {
// Recursion function
static void fun(int n)
{
if (n > 0)
{
System.out.print(n + " ");
// Last statement in the function
fun(n - 1);
}
}
// Driver Code
public static void main(String[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by pratham76.
# Code Showing Tail Recursion
# Recursion function
def fun(n):
if (n > 0):
print(n, end=" ")
# Last statement in the function
fun(n - 1)
# Driver Code
x = 3
fun(x)
# This code is contributed by Shubhamsingh10
// C# code Showing Tail Recursion
using System;
class GFG
{
// Recursion function
static void fun(int n)
{
if (n > 0)
{
Console.Write(n + " ");
// Last statement in the function
fun(n - 1);
}
}
// Driver Code
public static void Main(string[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by rutvik_56
<script>
// Javascript code Showing Tail Recursion
// Recursion function
function fun(n)
{
if (n > 0)
{
document.write(n + " ");
// Last statement in the function
fun(n - 1);
}
}
// Driver Code
var x = 3;
fun(x);
// This code is contributed by shivanisinghss2110
</script>
Let's understand the example by tracing tree of recursive function. That is how the calls are made and how the outputs are produced.
Time Complexity For Tail Recursion : O(n)
Space Complexity For Tail Recursion : O(n)
Note: Time & Space Complexity is given for this specific example. It may vary for another example.
Let's now converting Tail Recursion into Loop and compare each other in terms of Time & Space Complexity and decide which is more efficient.
C++
// Converting Tail Recursion into Loop
#include <iostream>
using namespace std;
void fun(int y)
{
while (y > 0) {
cout << y << " ";
y--;
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
//This Code is contributed by Shubhamsingh10
// Converting Tail Recursion into Loop
#include <stdio.h>
void fun(int y)
{
while (y > 0) {
printf("%d ", y);
y--;
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
// Converting Tail Recursion into Loop
import java.io.*;
class GFG
{
static void fun(int y)
{
while (y > 0) {
System.out.print(" "+ y);
y--;
}
}
// Driver code
public static void main(String[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by shivanisinghss2110
# Converting Tail Recursion into Loop
def fun(y):
while (y > 0):
print(y , end = " ")
y -= 1
# Driver code
x = 3
fun(x)
# This Code is contributed by shivanisinghss2110
// Converting Tail Recursion into Loop
using System;
class GFG
{
static void fun(int y)
{
while (y > 0) {
Console.Write(" "+ y);
y--;
}
}
// Driver code
public static void Main(String[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by shivanisinghss2110
<script>
function fun(y)
{
while (y > 0) {
document.write(" "+ y);
y--;
}
}
// Driver code
var x = 3;
fun(x);
// This code is contributed by shivanisinghss2110
</script>
Time Complexity: O(n)
Space Complexity: O(1)
Note: Time & Space Complexity is given for this specific example. It may vary for another example.
So it was seen that in case of loop the Space Complexity is O(1) so it was better to write code in loop instead of tail recursion in terms of Space Complexity which is more efficient than tail recursion.
Why space complexity is less in case of loop ?
Before explaining this I am assuming that you are familiar with the knowledge that's how the data stored in main memory during execution of a program. In brief,when the program executes,the main memory divided into three parts. One part for code section, the second one is heap memory and another one is stack memory. Remember that the program can directly access only the stack memory, it can't directly access the heap memory so we need the help of pointer to access the heap memory.
Let's now understand why space complexity is less in case of loop ?
In case of loop when function "(void fun(int y))" executes there only one activation record created in stack memory(activation record created for only 'y' variable) so it takes only 'one' unit of memory inside stack so it's space complexity is O(1) but in case of recursive function every time it calls itself for each call a separate activation record created in stack.So if there's 'n' no of call then it takes 'n' unit of memory inside stack so it's space complexity is O(n).
- Head Recursion: If a recursive function calling itself and that recursive call is the first statement in the function then it's known as Head Recursion. There's no statement, no operation before the call. The function doesn't have to process or perform any operation at the time of calling and all operations are done at returning time.
Example:
C++
// C++ program showing Head Recursion
#include <bits/stdc++.h>
using namespace std;
// Recursive function
void fun(int n)
{
if (n > 0) {
// First statement in the function
fun(n - 1);
cout << " "<< n;
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
// this code is contributed by shivanisinghss2110
// C program showing Head Recursion
#include <stdio.h>
// Recursive function
void fun(int n)
{
if (n > 0) {
// First statement in the function
fun(n - 1);
printf("%d ", n);
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
// Java program showing Head Recursion
import java.io.*;
class GFG{
// Recursive function
static void fun(int n)
{
if (n > 0) {
// First statement in the function
fun(n - 1);
System.out.print(" "+ n);
}
}
// Driver code
public static void main(String[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by shivanisinghss2110
# Python program showing Head Recursion
# Recursive function
def fun(n):
if (n > 0):
# First statement in the function
fun(n - 1)
print(n,end=" ")
# Driver code
x = 3
fun(x)
# this code is contributed by shivanisinghss2110
// Java program showing Head Recursion
using System;
class GFG{
// Recursive function
static void fun(int n)
{
if (n > 0) {
// First statement in the function
fun(n - 1);
Console.Write(" "+ n);
}
}
// Driver code
public static void Main(String[] args)
{
int x = 3;
fun(x);
}
}
// This code is contributed by shivanisinghss2110
<script>
// JavaScript program showing Head Recursion
// Recursive function
function fun(n)
{
if (n > 0) {
// First statement in the function
fun(n - 1);
document.write(" "+ n);
}
}
// Driver code
var x = 3;
fun(x);
// This code is contributed by shivanisinghss2110
</script>
Let's understand the example by tracing tree of recursive function. That is how the calls are made and how the outputs are produced.
Time Complexity For Head Recursion: O(n)
Space Complexity For Head Recursion: O(n)
Note: Time & Space Complexity is given for this specific example. It may vary for another example.
Note: Head recursion can't easily convert into loop as Tail Recursion but it can. Let's convert the above code into the loop.
C++
// Converting Head Recursion into Loop
#include <iostream>
using namespace std;
// Recursive function
void fun(int n)
{
int i = 1;
while (i <= n) {
cout <<" "<< i;
i++;
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
// this code is contributed by shivanisinghss2110
// Converting Head Recursion into Loop
#include <stdio.h>
// Recursive function
void fun(int n)
{
int i = 1;
while (i <= n) {
printf("%d ", i);
i++;
}
}
// Driver code
int main()
{
int x = 3;
fun(x);
return 0;
}
// Converting Head Recursion into Loop
import java.util.*;
class GFG
{
// Recursive function
static void fun(int n)
{
int i = 1;
while (i <= n) {
System.out.print(" "+ i);
i++;
}
}
// Driver code
public static void main(String[] args)
{
int x = 3;
fun(x);
}
}
// this code is contributed by shivanisinghss2110
# Converting Head Recursion into Loop
# Recursive function
def fun(n):
i = 1
while (i <= n):
print(i,end=" ")
i+=1
# Driver code
x = 3
fun(x)
# This code is contributed by shivanisinghss2110
// Converting Head Recursion into Loop
using System;
class GFG
{
// Recursive function
static void fun(int n)
{
int i = 1;
while (i <= n) {
Console.Write(" "+ i);
i++;
}
}
// Driver code
public static void Main(String[] args)
{
int x = 3;
fun(x);
}
}
// this code is contributed by shivanisinghss2110
<script>
// Converting Head Recursion into Loop
// Recursive function
function fun(n)
{
var i = 1;
while (i <= n) {
document.write(" "+ i);
i++;
}
}
// Driver code
var x = 3;
fun(x);
// this code is contributed by shivanisinghss2110
</script>
- Tree Recursion: To understand Tree Recursion let's first understand Linear Recursion. If a recursive function calling itself for one time then it's known as Linear Recursion. Otherwise if a recursive function calling itself for more than one time then it's known as Tree Recursion.
Example:
Pseudo Code for linear recursion
fun(n)
{
// some code
if(n>0)
{
fun(n-1); // Calling itself only once
}
// some code
}
Program for tree recursion
C++
// C++ program to show Tree Recursion
#include <iostream>
using namespace std;
// Recursive function
void fun(int n)
{
if (n > 0)
{
cout << " " << n;
// Calling once
fun(n - 1);
// Calling twice
fun(n - 1);
}
}
// Driver code
int main()
{
fun(3);
return 0;
}
// This code is contributed by shivanisinghss2110
// C program to show Tree Recursion
#include <stdio.h>
// Recursive function
void fun(int n)
{
if (n > 0) {
printf("%d ", n);
// Calling once
fun(n - 1);
// Calling twice
fun(n - 1);
}
}
// Driver code
int main()
{
fun(3);
return 0;
}
// Java program to show Tree Recursion
class GFG
{
// Recursive function
static void fun(int n)
{
if (n > 0) {
System.out.print(" "+ n);
// Calling once
fun(n - 1);
// Calling twice
fun(n - 1);
}
}
// Driver code
public static void main(String[] args)
{
fun(3);
}
}
// This code is contributed by shivanisinghss2110
# C++ program to show Tree Recursion
# Recursive function
def fun(n):
if (n > 0):
print(n, end=" ")
# Calling once
fun(n - 1)
# Calling twice
fun(n - 1)
# Driver code
fun(3)
# This code is contributed by shivanisinghss2110
// C# program to show Tree Recursion
using System;
class GFG
{
// Recursive function
static void fun(int n)
{
if (n > 0) {
Console.Write(" "+ n);
// Calling once
fun(n - 1);
// Calling twice
fun(n - 1);
}
}
// Driver code
public static void Main(String[] args)
{
fun(3);
}
}
// This code is contributed by shivanisinghss2110
<script>
// JavaScript program to show Tree Recursion
// Recursive function
function fun(n)
{
if (n > 0) {
document.write(" "+ n);
// Calling once
fun(n - 1);
// Calling twice
fun(n - 1);
}
}
// Driver code
fun(3);
// This code is contributed by shivanisinghss2110
</script>
Let's understand the example by tracing tree of recursive function. That is how the calls are made and how the outputs are produced.
Time Complexity For Tree Recursion: O(2^n)
Space Complexity For Tree Recursion: O(n)
Note: Time & Space Complexity is given for this specific example. It may vary for another example.
- Nested Recursion: In this recursion, a recursive function will pass the parameter as a recursive call. That means "recursion inside recursion". Let see the example to understand this recursion.
Example:
C++
// C++ program to show Nested Recursion
#include <iostream>
using namespace std;
int fun(int n)
{
if (n > 100)
return n - 10;
// A recursive function passing parameter
// as a recursive call or recursion inside
// the recursion
return fun(fun(n + 11));
}
// Driver code
int main()
{
int r;
r = fun(95);
cout << " " << r;
return 0;
}
// This code is contributed by shivanisinghss2110
// C program to show Nested Recursion
#include <stdio.h>
int fun(int n)
{
if (n > 100)
return n - 10;
// A recursive function passing parameter
// as a recursive call or recursion
// inside the recursion
return fun(fun(n + 11));
}
// Driver code
int main()
{
int r;
r = fun(95);
printf("%d\n", r);
return 0;
}
// Java program to show Nested Recursion
import java.util.*;
class GFG {
static int fun(int n)
{
if (n > 100)
return n - 10;
// A recursive function passing parameter
// as a recursive call or recursion
// inside the recursion
return fun(fun(n + 11));
}
// Driver code
public static void main(String args[])
{
int r;
r = fun(95);
System.out.print(" "+ r);
}
}
// This code is contributed by shivanisinghss2110
# Python program to show Nested Recursion
def fun(n):
if (n > 100):
return n - 10
# A recursive function passing parameter
# as a recursive call or recursion inside
# the recursion
return fun(fun(n + 11))
# Driver code
r = fun(95)
print("", r)
# This code is contributed by shivanisinghss2110
// C# program to show Nested Recursion
using System;
class GFG {
static int fun(int n)
{
if (n > 100)
return n - 10;
// A recursive function passing parameter
// as a recursive call or recursion
// inside the recursion
return fun(fun(n + 11));
}
// Driver code
public static void Main(String []args)
{
int r;
r = fun(95);
Console.Write(" "+ r);
}
}
// This code is contributed by shivanisinghss2110
<script>
// JavaScript program to show Nested Recursion
function fun( n)
{
if (n > 100)
return n - 10;
// A recursive function passing parameter
// as a recursive call or recursion
// inside the recursion
return fun(fun(n + 11));
}
// Driver code
var r;
r = fun(95);
document.write(" "+ r);
// This code is contributed by shivanisinghss2110
</script>
Let's understand the example by tracing tree of recursive function. That is how the calls are made and how the outputs are produced.
2. Indirect Recursion: In this recursion, there may be more than one functions and they are calling one another in a circular manner.
From the above diagram fun(A) is calling for fun(B), fun(B) is calling for fun(C) and fun(C) is calling for fun(A) and thus it makes a cycle.
Example:
C++
// C++ program to show Indirect Recursion
#include <iostream>
using namespace std;
void funB(int n);
void funA(int n)
{
if (n > 0) {
cout <<" "<< n;
// fun(A) is calling fun(B)
funB(n - 1);
}
}
void funB(int n)
{
if (n > 1) {
cout <<" "<< n;
// fun(B) is calling fun(A)
funA(n / 2);
}
}
// Driver code
int main()
{
funA(20);
return 0;
}
// this code is contributed by shivanisinghss2110
// C program to show Indirect Recursion
#include <stdio.h>
void funB(int n);
void funA(int n)
{
if (n > 0) {
printf("%d ", n);
// Fun(A) is calling fun(B)
funB(n - 1);
}
}
void funB(int n)
{
if (n > 1) {
printf("%d ", n);
// Fun(B) is calling fun(A)
funA(n / 2);
}
}
// Driver code
int main()
{
funA(20);
return 0;
}
// Java program to show Indirect Recursion
import java.io.*;
class GFG{
static void funA(int n)
{
if (n > 0) {
System.out.print(" " +n);
// Fun(A) is calling fun(B)
funB(n - 1);
}
}
static void funB(int n)
{
if (n > 1) {
System.out.print(" " +n);
// Fun(B) is calling fun(A)
funA(n / 2);
}
}
// Driver code
public static void main (String[] args)
{
funA(20);
}
}
// This code is contributed by shivanisinghss2110
// C# program to show Indirect Recursion
using System;
class GFG{
static void funA(int n)
{
if (n > 0) {
Console.Write(" " +n);
// Fun(A) is calling fun(B)
funB(n - 1);
}
}
static void funB(int n)
{
if (n > 1) {
Console.Write(" " +n);
// Fun(B) is calling fun(A)
funA(n / 2);
}
}
// Driver code
public static void Main (String[] args)
{
funA(20);
}
}
// This code is contributed by shivanisinghss2110
# Python program to show Indirect Recursion
def funA(n):
if (n > 0):
print("", n, end='')
# Fun(A) is calling fun(B)
funB(n - 1)
def funB( n):
if (n > 1):
print("", n, end='')
# Fun(B) is calling fun(A)
funA(n // 2)
# Driver code
funA(20)
# This code is contributed by shivanisinghss2110
<script>
// JavaScript program to show Indirect Recursion
function funA(n)
{
if (n > 0) {
document.write(n.toFixed(0) + "</br>");
// Fun(A) is calling fun(B)
funB(n - 1);
}
}
function funB(n)
{
if (n > 1) {
document.write(n.toFixed(0) + "</br>");
// Fun(B) is calling fun(A)
funA(n / 2);
}
}
// Driver code
funA(20);
// this code is contributed by shivanisinghss2110
</script>
Let's understand the example by tracing tree of recursive function. That is how the calls are made and how the outputs are produced.
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