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 recursive fibonacci code. 
 

// 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;
}

// 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) ); 
    }
}

# 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# 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
// 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.
?>

<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)