Why does the fibonacci recursive sequence work? - function

Question

I am wondering why this Fibonacci recursive function works:
int fibRec(int n)
{
if ((n == 1) || (n == 0))
{
return n;
}
int i = fibRec(n - 1) + fibRec(n - 2);
return i;
}
I understand what the Fibonacci sequence is and I understand what a recursive function does and how this function is working. I'm just having troubles understanding why it works. I know that when you break it down, you are essentially adding a bunch of 0s and 1s, as this image depicts.
fibonacci recursive
But why is it that when I pass a 5 to the function and all the 0 and 1s are added that it will equal the 5th sequence number in the Fibonacci sequence? I've seen this question asked before but never really explained. The responses are all just "because recursion". Yes, I know what a recursive function is and how this one is working. But WHY does this recursive function give you the correct Fibonacci sequence number?

Answer 1

In the Fibonacci sequence the first two numbers are zero and one. Every number after these is the sum of the previous 2 numbers. So the first few numbers are
F(0) ≡ 0
F(1) ≡ 1
F(2) = F(1) + F(0) = 1 + 0 = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(5) + F(4) = 5 + 3 = 8
...
F(n) = F(n - 1) + F(n - 2) ∀ n > 1
Therefore when we calculate a Fibonacci number recursively we have to practice the following logical procedure (in pseudo-code out of respect to StackOverflow).
Integer NthFibonacci(Integer n) {
if (n < 0) {
return undefined;
} else if (n < 2) {
return n;
} else {
return NthFibonacci(n - 1) + NthFibonacci(n - 2);
}
}
I'm sure you know all this but I think it will help my explanation to have this part as a reference.
Where the Ones and Zeros Come In
The best way to explain this is probably with an example.
Imagine that, as above, we are trying to recursively calculate F(6). Try following the procedure given above. Remember that we will perform recursion only if n > 1.
First we start with F(6) = F(5) + F(4).
Then we find F(5) = F(4) + F(3).
Then we find F(4) = F(3) + F(2).
Then we find F(3) = F(2) + F(1).
Then we find F(2) = F(1) + F(0).
This is where things start to work out!
We have now gotten F(2) in terms of F(1) ≡ 1 and F(0) ≡ 0 (both of which are known), and so we are able to calculate an actual value instead of performing more recursion.
We can now find F(2) = F(1) + F(0) = 1 + 0 = 1.
NOTICE THE 1 AND 0 Those are what people are talking about when they say the whole thing comes down to ones and zeros. Every time we recurse down to find a base value we will end up finding F(2) = 1 + 0. This leads to more ones and zeros as we move back up our recursion tree being able to calculate higher and higher values, as follows.
F(3) = F(2) + F(1) = (1 + 0) + 1
F(4) = F(3) + F(2) = ((1 + 0) + 1) + (1 + 0)
F(5) = F(4) + F(3) = (((1 + 0) + 1) + (1 + 0)) + ((1 + 0) + 1)
F(6) = F(5) + F(4) = ((((1 + 0) + 1) + (1 + 0)) + ((1 + 0) + 1)) + (((1 + 0) + 1) + (1 + 0))
Now if you add up all the 1's you get a sum of 8, and so F(6) = 8, which is correct!
This is how it works, and this is how it breaks down to ones and zeros.

Answer 2

Remember, recursion works by breaking down the problem till we know what the answer is, and then building it up from there.
What do we know about the fibonacci sequence?
We know that when:
x = 1
and
x = 0
That that is the lowest it goes. That is an important key. Because when x = 0 we are really doing 0 + 0 and when x = 1 we are really doing 0 + 1. Now start at the top.
0,1,1,2,3,5,8,13...
If we are at 13. what is 13? Why simply 5 + 8 right? So That is where
int i = fibRec(n - 1) + fibRec(n - 2);
comes from. Because these are going to branch out lower and lower till we are at a base case for each one.
This is the recursive calling. Because now the method is going to go back to the stack and call fibRec again. You will notice that (n-1) and (n-2) are both added together and set to i. This is so that we don't lose the value. because of the + sign the stack then ends up returning more and more (n-1)s and (n-2)s until we are at the base case. I hope all of this makes sense. Thinking recursively can be very difficult. Here is a a visual representation from top to bottom of what it would look like.
In short. This just keeps adding the previous fibonacci sequences to the current one until it gets to the current loop.