How does this recursive function in Python work?

Question

I am learning python from the Book: "ThinkPython".

At page 56 (Chapter 6. Fruitful functions) there is a recursive function that calculates the factorial of any number. It does work, however I don't understand why. This is the code:

def factorial(n):
    if n == 0:
        return 1
    else:
        recurse = factorial(n-1)
        result = n * recurse
        return result

Let's say I try with 3, I guess this is what should happen:

1. enters the factorial function and n=3
2. enters the else statement because n is not 0
3. and here goes back to the beginning to step 1 with n = 2

so it should do the same until n = 0 and return 1 (it will never get to the last lines).

Answer 1

Your function is:

def factorial(n):
 if n == 0:
  return 1
 else:
  recurse = factorial(n-1)
  result = n * recurse
  return result

Let's go through an execution of this function step-by-step, starting with n = 3.

Entering 
STATE 1 
n = 3 
recurse = factorial(3-1)

STATE 2
n = 2
recurse = factorial(2-1)

STATE 3
n = 1
recurse = factorial(1-1)

STATE 4
You now have an n = 0, so you return 1 to STATE 3.

STATE 3
n = 1
recurse = 1
result = 1
return result to STATE 2

STATE 2
n = 2
recurse = 1
result = 2 * 1 = 2
return result to STATE 1

STATE 1
n = 3
recurse = 2
result = 3 * 2 = 6

And then 6, the factorial of 3, is returned to the calling function :D

Recursion really is a tricky thing to confront, it just takes time to get used to it and to the different ways that it can be used.

Answer 2

You're correct, as far as you go. However, you don't continue far enough. When the factorial(0) call returns, where does it return to? it returns to the

recurse = factorial (n - 1)

line where n = 1 , and then you continue on from there. So, recurse = 1 , result = 1* 1 = 1 , and returns 1. Again, that goes to the recurse = line in the n=2 case.so, result = 2*1 = 2 , and returns that, to the recurse = line where n = 3, so result = 2 * 3 = 6 , which is what gets returned.

Answer 3

The factorial of n is defined as n * (n-1) * (n-2) etc until 1

So for n = 5 -> 5*4*3*2*1

In the code the factorial is defined as n * whatever the factorial of n-1 is

Python will find out this last part by asking itself: what is the factorial of n-1 ? Well it is (n-1) * whatever the factorial of (n-1)-1 is.

Do you see the loop/ recursion?

Since for n==0 the factorial will return 1 , python can ask itself over and over again what the recursive part is, until it asks itself, what is the factorial of 0? . This part is hardcoded in the first part.

Once it enters this last part, it knows the other answers too! For n == 5 :

• The factorial is 5 * (factorial of 4)
  • The factorial is 4 * (factorial of 3)
    • ...
      • The factorial of 0 is 1.

The factorial of 5 = 5*4*3*2*1