# 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.

## 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` `) );` ` ` `}` `}` |

## Python3

`# 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.` `?>` |

## 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)

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