What Is Recursion and How Does It Work

What is recursion

Basically, a function is recursive when

1. A function has a simple base case, and when
2. All other cases have rules which reduce to the base case.

As an example, to calculate an factorial:

public static long factorial(int i)
{
    // This is the base case
    if(i == 0)
    {
         return 1;
    }
    else
    {
        // This reduces the problem to something closer to the base case
        return i * factorial(i - 1);
    }
}

What is recursion and how does it work?

A recursive function/method calls itself. For a recursive algorithm to terminate you need a base case (e.g. a condition where the function does not call itself recursively) and you also need to make sure that you get closer to that base case in each recursive call. Let's look at a very simple example:

def countdown(n)
  return if n.zero? # base case
  puts n
  countdown(n-1)    # getting closer to base case 
end               

countdown(5)
5
4
3
2
1

Some problems can be very elegantly expressed with recursion, e.g a lot of mathematical functions are described in a recursive way.

Understanding how recursive functions work

I think the confusion is stemming from thinking of it as "the same function" being called many times. If you think of it as "many copies of the same function being called", then it may be clearer:

Only one copy of the function ever returns 0, and it's not the first one (it's the last one). So the result of calling the first one is not 0.

For the second bit of confusion, I think it will be easier to spell out the recursion in English. Read this line:

return a + sumInts(a + 1, b: b)

as "return the value of 'a' plus (the return value of another copy of the function, which is the copy's value of 'a' plus (the return value of another copy of the function, which is the second copy's value of 'a' plus (...", with each copy of the function spawning a new copy of itself with a increased by 1, until the a > b condition is met.

By the time you reach the the a > b condition being true, you have a (potentially arbitrarily) long stack of copies of the function all in the middle of being run, all waiting on the result of the next copy to find out what they should add to 'a'.

(edit: also, something to be aware of is that the stack of copies of the function I mention is a real thing that takes up real memory, and will crash your program if it gets too large. The compiler can optimize it out in some cases, but exhausting stack space is a significant and unfortunate limitation of recursive functions in many languages)

What is tail recursion?

Consider a simple function that adds the first N natural numbers. (e.g. sum(5) = 0 + 1 + 2 + 3 + 4 + 5 = 15).

Here is a simple JavaScript implementation that uses recursion:

function recsum(x) {
    if (x === 0) {
        return 0;
    } else {
        return x + recsum(x - 1);
    }
}

If you called recsum(5), this is what the JavaScript interpreter would evaluate:

recsum(5)
5 + recsum(4)
5 + (4 + recsum(3))
5 + (4 + (3 + recsum(2)))
5 + (4 + (3 + (2 + recsum(1))))
5 + (4 + (3 + (2 + (1 + recsum(0)))))
5 + (4 + (3 + (2 + (1 + 0))))
5 + (4 + (3 + (2 + 1)))
5 + (4 + (3 + 3))
5 + (4 + 6)
5 + 10
15

Note how every recursive call has to complete before the JavaScript interpreter begins to actually do the work of calculating the sum.

Here's a tail-recursive version of the same function:

function tailrecsum(x, running_total = 0) {
    if (x === 0) {
        return running_total;
    } else {
        return tailrecsum(x - 1, running_total + x);
    }
}

Here's the sequence of events that would occur if you called tailrecsum(5), (which would effectively be tailrecsum(5, 0), because of the default second argument).

tailrecsum(5, 0)
tailrecsum(4, 5)
tailrecsum(3, 9)
tailrecsum(2, 12)
tailrecsum(1, 14)
tailrecsum(0, 15)
15

In the tail-recursive case, with each evaluation of the recursive call, the running_total is updated.

Note: The original answer used examples from Python. These have been changed to JavaScript, since Python interpreters don't support tail call optimization. However, while tail call optimization is part of the ECMAScript 2015 spec, most JavaScript interpreters don't support it.

How does recursion work?

Here's a simple example of recursive method: -

public int recur(int count) {
   if (count < 10) {
       return count + recur(count++);
   }
   return count;
}

System.out.println(recur(0));  // Invoke first time

Understanding recursion in Python

lets walk through the execution.

fact(5):
   5 is not 0, so fact(5) = 5 * fact(4)
   what is fact(4)?
fact(4):
   4 is not 0, so fact(4) = 4 * fact(3)
   what is fact(3)?
fact(3):
   3 is not 0, so fact(3) = 3 * fact(2)
   what is fact(2)?
fact(2):
   2 is not 0, so fact(2) = 2 * fact(1)
   what is fact(1)?
fact(1):
   1 is not 0, so fact(1) = 1 * fact(0)
   what is fact(0)?
fact(0):
   0 IS 0, so fact(0) is 1

Now lets gather our result.

fact(5) = 5* fact(4)

substitute in our result for fact(4)

fact(5) = 5 * 4 * fact(3)

substitute in our result for fact(3)

fact(5) = 5 * 4 * 3 * fact(2)

substitute in our result for fact(2)

fact(5) = 5 * 4 * 3 * 2 * fact(1)

substitute in our result for fact(1)

fact(5) = 5 * 4 * 3 * 2 * 1 * fact(0)

substitute in our result for fact(0)

fact(5) = 5 * 4 * 3 * 2 * 1 * 1 = 120

And there you have it. Recursion is the process of breaking a larger problem down by looking at it as successfully smaller problems until you reach a trivial (or "base") case.

Recursive function in simple english

Three words - It Calls Itself.

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