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. 
 

Output : 
 

fib(9) = 34

Analysis of Algorithm 
 

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