Recursion termination

In computing, recursion termination is when certain conditions are met and a recursive algorithm stops calling itself and begins to return values. This happens only if, with every recursive call, the recursive algorithm changes its state and moves toward the base case. Cases that satisfy the definition, without being defined in terms of that definition, are called base cases. They are small enough to solve directly.

Examples

Fibonacci Function

The Fibonacci function(fibonacci(n)), which takes integer n(n >= 0) as input, has three conditions

1. If n is 0, returns 0.
2. If n is 1, returns 1.
3. Otherwise, return [fibonacci(n-1) + fibonacci(n-2)]

This recursive function terminates if either conditions 1 or 2 are satisfied. We see that the function's recursive call reduces the value of n(by passing n-1 or n-2 in the function) ensuring that n reaches either condition 1 or 2.

If we look at an example of this in Python we can see what's going on a bit more clearly:

def fibonacci (n):
    if n == 1:
        return 1
    elif n == 2:
        return 1
    elif n > 2:
        return fibonacci(n-1) + fibonacci(n-2)

Factorial Example

An example in the programming language C++:

int factorial(int number)  
{
	if (number == 0)
		return 1;
	else
		return (number * factorial(number - 1)); 
}

Here we see that in the recursive call, the number passed in the recursive step is reduced by 1. This again ensures that the number will at some point reduce to 0 which in turn terminates the recursive algorithm since we have our base case where we know that the factorial of 0 is 1 (or 0! = 1).

In the C programing language we could similarly do something like this:

long factorial(int n)
{
  if (n == 0)
    return 1;
  else
    return(n * factorial(n-1));
}

External links

• Princeton university: "An introduction to computer science in the context of scientific, engineering, and commercial applications"
• University of Toronto: "Introduction to Computer Science"
• Computerphile: "What On Earth is Recursion?"