recursion - How does the fibonacci recursive function "work"?

Question

I'm new to Javascript and was reading up on it, when I came to a chapter that described function recursion. It used an example function to find the nth number of the Fibonacci sequence. The code is as follows:

function fibonacci(n) {
    if (n < 2){
        return 1;
    }else{
        return fibonacci(n-2) + fibonacci(n-1);
    }
}

console.log(fibonacci(7));
//Returns 21

I'm having trouble grasping exactly what this function is doing. Can someone explain what's going on here? I'm getting stuck on the 5th line, where the function calls itself. What's happening here?

Answer 1

You're defining a function in terms of itself. In general, fibonnaci(n) = fibonnaci(n - 2) + fibonnaci(n - 1). We're just representing this relationship in code. So, for fibonnaci(7) we can observe:

• fibonacci(7) is equal to fibonacci(6) + fibonacci(5)
• fibonacci(6) is equal to fibonacci(5) + fibonacci(4)
• fibonacci(5) is equal to fibonacci(4) + fibonacci(3)
• fibonacci(4) is equal to fibonacci(3) + fibonacci(2)
• fibonacci(3) is equal to fibonacci(2) + fibonacci(1)
• fibonacci(2) is equal to fibonacci(1) + fibonacci(0)
• fibonacci(1) is equal to 1
• fibonacci(0) is equal to 1

We now have all the parts needed to evaluate fibonacci(7), which was our original goal. Notice that the base case -- return 1 when n < 2 -- is what makes this possible. This is what stops the recursion, so that we can can start the process of unrolling the stack and summing the values we're returning at each step. Without this step, we'd continue calling fibonacci on smaller and smaller values right up until the program finally crashed.

It might help to add some logging statements that illustrate this:

function fibonacci(n, c) {
    var indent = "";
    for (var i = 0; i < c; i++) {
        indent += " ";
    }
    console.log(indent + "fibonacci(" + n + ")");
    if (n < 2) {
        return 1;
    } else {
        return fibonacci(n - 2, c + 4) + fibonacci(n - 1, c + 4);
    }
}

console.log(fibonacci(7, 0));

Output:

fibonacci(7)
    fibonacci(5)
        fibonacci(3)
            fibonacci(1)
            fibonacci(2)
                fibonacci(0)
                fibonacci(1)
        fibonacci(4)
            fibonacci(2)
                fibonacci(0)
                fibonacci(1)
            fibonacci(3)
                fibonacci(1)
                fibonacci(2)
                    fibonacci(0)
                    fibonacci(1)
    fibonacci(6)
        fibonacci(4)
            fibonacci(2)
                fibonacci(0)
                fibonacci(1)
            fibonacci(3)
                fibonacci(1)
                fibonacci(2)
                    fibonacci(0)
                    fibonacci(1)
        fibonacci(5)
            fibonacci(3)
                fibonacci(1)
                fibonacci(2)
                    fibonacci(0)
                    fibonacci(1)
            fibonacci(4)
                fibonacci(2)
                    fibonacci(0)
                    fibonacci(1)
                fibonacci(3)
                    fibonacci(1)
                    fibonacci(2)
                        fibonacci(0)
                        fibonacci(1)

Values at the same level of indentation are summed to produce the result for the previous level of indentation.