Tail Recursion for Fibonacci

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++

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// Tail Recursive Fibonacci
// implementation
#include <iostream>
using namespace std;
  
// A tail recursive function to
// calculate n th fibnacci 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;
}

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Java














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// Tail Recursive  


// Fibonacci implementation 


  


class GFG 


{ 


    // A tail recursive function to 


    // calculate n th fibnacci 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) );  


    } 


} 



















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Python














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# A tail recursive function to  


# calculate n th fibnacci 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))) 



















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C#














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// C# Program for Tail 


// Recursive Fibonacci  


using System; 


  


class GFG 


{ 


      


    // A tail recursive function to 


    // calculate n th fibnacci 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. 



















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PHP














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<?php 


// A tail recursive PHP function to 


// calculate n th fibnacci 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. 


?> 



















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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 contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.


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