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Tail Recursion for Fibonacci

  • Difficulty Level : Easy
  • Last Updated : 07 Jul, 2021

Write a tail recursive function for calculating the n-th Fibonacci number. 
Examples : 
 

Input : n = 4
Output : fib(4) = 3

Input : n = 9
Output : fib(9) = 34

Prerequisites : Tail Recursion, Fibonacci numbers
A recursive function is tail recursive when the recursive call is the last thing executed by the function. 
 

Writing a tail recursion is little tricky. To get the correct intuition, we first look at the iterative approach of calculating the n-th Fibonacci number. 
 

int fib(int n)
{
  int a = 0, b = 1, c, i;
  if (n == 0)
    return a;
  for (i = 2; i <= n; i++)
  {
     c = a + b;
     a = b;
     b = c;
  }
  return b;
}

Here there are three possibilities related to n :- 
 

n == 0

 



n == 1

 

n > 1

First two are trivial. We focus on discussion of the case when n > 1. 
In our iterative approach for n > 1, 
We start with 
 

a = 0
b = 1

For n-1 times we repeat following for ordered pair (a,b) 
Though we used c in actual iterative approach, but the main aim was as below :- 
 

(a, b) = (b, a+b)

We finally return b after n-1 iterations.
Hence we repeat the same thing this time with the recursive approach. We set the default values 
 

a = 0
b = 1

Here we’ll recursively call the same function n-1 times and correspondingly change the values of a and b. 
Finally, return b.
If its case of n == 0 OR n == 1, we need not worry much!
Here is implementation of tail recurssive fibonacci code. 
 

C++




// Tail Recursive Fibonacci
// implementation
#include <iostream>
using namespace std;
 
// A tail recursive function to
// calculate n th fibonacci number
int fib(int n, int a = 0, int b = 1)
{
    if (n == 0)
        return a;
    if (n == 1)
        return b;
    return fib(n - 1, b, a + b);
}
 
// Driver Code
int main()
{
    int n = 9;
    cout << "fib(" << n << ") = "
         << fib(n) << endl;
    return 0;
}

Java




// Tail Recursive
// Fibonacci implementation
 
class GFG
{
    // A tail recursive function to
    // calculate n th fibonacci number
    static int fib(int n, int a, int b )
    {
         
        if (n == 0)
            return a;
        if (n == 1)
            return b;
        return fib(n - 1, b, a + b);
    }
     
    public static void main (String[] args)
    {
        int n = 9;
        System.out.println("fib(" + n +") = "+
                                 fib(n,0,1) );
    }
}

Python




# A tail recursive function to
# calculate n th fibonacci number
def fib(n, a = 0, b = 1):
    if n == 0:
        return a
    if n == 1:
        return b
    return fib(n - 1, b, a + b);
 
# Driver Code
n = 9;
print("fib("+str(n)+") = "+str(fib(n)))

C#




// C# Program for Tail
// Recursive Fibonacci
using System;
 
class GFG
{
     
    // A tail recursive function to
    // calculate n th fibonacci number
    static int fib(int n, int a , int b )
    {
        if (n == 0)
            return a;
        if (n == 1)
            return b;
        return fib(n - 1, b, a + b);
    }
     
    // Driver Code
    public static void Main ()
    {
        int n = 9;
        Console.Write("fib(" + n +") = " +
                           fib(n, 0, 1) );
    }
}
 
// This code is contributed
// by nitin mittal.

PHP




<?php
// A tail recursive PHP function to
// calculate n th fibonacci number
function fib($n, $a = 0, $b = 1)
{
    if ($n == 0)
        return $a;
    if ($n == 1)
        return $b;
    return fib($n - 1, $b, $a + $b);
}
 
// Driver Code
$n = 9;
echo "fib($n) = " , fib($n);
return 0;
 
// This code is contributed by nitin mittal.
?>

C++




#include <iostream>
using namespace std;
 
int main() {
 
    cout<<"GFG!";
    return 0;
}

Javascript




<script>
 
// A tail recursive Javascript function to
// calculate n th fibonacci number
 
function fib(n, a = 0, b = 1)
{
    if (n == 0){
        return a;
    }
    if (n == 1){
        return b;
    }
    return fib(n - 1, b, a + b);
}
 
// Driver Code
let n = 9;
document.write(`fib(${n}) =  ${fib(n)}`);
 
 
// This code is contributed by _saurabh_jaiswal.
 
</script>

Output : 
 

fib(9) = 34

Analysis of Algorithm 
 

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

This article is contributed by Pratik Chhajer. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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