The Fibonacci numbers are the numbers in the following integer sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ……..

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

F_{n}= F_{n-1}+ F_{n-2}

with seed values

F_{0}= 0 and F_{1}= 1.

Write a function *int fib(int n)* that returns F_{n}. For example, if *n* = 0, then *fib()* should return 0. If n = 1, then it should return 1. For n > 1, it should return F_{n-1} + F_{n-2}

For n = 9 Output:34

Following are different methods to get the nth Fibonacci number.

**Method 1 ( Use recursion ) **

A simple method that is a direct recursive implementation mathematical recurrence relation given above.

Output

34

*Time Complexity:* T(n) = T(n-1) + T(n-2) which is exponential.

We can observe that this implementation does a lot of repeated work (see the following recursion tree). So this is a bad implementation for nth Fibonacci number.

fib(5) / \ fib(4) fib(3) / \ / \ fib(3) fib(2) fib(2) fib(1) / \ / \ / \ fib(2) fib(1) fib(1) fib(0) fib(1) fib(0) / \ fib(1) fib(0)

*Extra Space:* O(n) if we consider the function call stack size, otherwise O(1).

**Method 2 ( Use Dynamic Programming )**

We can avoid the repeated work done is the method 1 by storing the Fibonacci numbers calculated so far.

Output:

34

*Time Complexity:* O(n)

*Extra Space: *O(n)

**Method 3 ( Space Optimized Method 2 )**

We can optimize the space used in method 2 by storing the previous two numbers only because that is all we need to get the next Fibonacci number in series.

Output:

*Time Complexity:* O(n)

*Extra Space: *O(1)

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