Flow of a factorial function using recursion

Question

I'm learning the fundamentals of Python right now by reading Thinking Python.

As I was reading a chapter about "fruitful" function, that is, a function that "returns" a value, I came across this problem. I can more or less figure out how the code of the factorial function works, but not exactly. I have trouble in understanding precisely the flow of a function involving self-calling or recursion. Btw, the printer statments in the code shown as below are scaffolding, a device used to track the flow of the function.

Without further ado, here is the code.

def factorial(n):
    space=' '*(2*n)
    print(space, 'factorial', n)
    if n==0:
        print(space, 'returning 1')
        return 1
    else:
        recurse = factorial(n-1)
        result=n*recurse
        print(space, 'Returning', result)
        return result

factorial(3) 

The result or the output of this code is shockingly a v-shape. I, for the life of me, cannot figure out why the computer (or the compiler?) runs the program the way it does here. It seems like it firstly prints the first line and the last line, and then goes on to print the second line and the second to the last line...This is strange. You would expect that it always goes top down.

To illustrate the oddity of this issue, I hereby attach another piece of code that prints numbers as you count down. The code is similar and simpler, and you would expect it to behave similarly to the factorial function before. But it doesn't. There is no oddity in the order in which the lines of outcome are displayed or printed off.

def f(n):
    print(n)
    if n==0:
        return 0
    else:
        print(3*n)
        f(n-1)
        return

I would really appreciate it if you could help me with this!

Answer 1

Here's your code a bit restructured to make it easier to see:

def factorial(n):
    space = ' '*(2*n)
    print(space, 'factorial', n)
    result = 1 if n == 0 else n*factorial(n-1)        
    print(space, 'Returning', result)
    return result

And that's the output for factorial(3) with the "V shape" you mentioned:

       factorial 3
     factorial 2
   factorial 1
 factorial 0
 Returning 1
   Returning 1
     Returning 2
       Returning 6

It looks correct and does the job of calculating the factorial. To understand the order of prints seen here, think of the following:

• The prints start with the call for 3, since that's your entry point
• The indentation is then 3x2 = 6 spaces
• Since for the calculation of factorial(3) the reuslt of factioral(2) is needed, this call comes next, with an indentation of 2x2 = 4 spaces. And so on for all further factorials. That's where the V shape of the first half of prints comes from.
• All of these function calls are now "waiting" for a result to return, at the position of the recursive call in the code. Since only factorial(0) can deliver that return value without another recursion, it will print the first return value. With an indentation of 0.
• With the above return value, the function call to factorial(1) can now calculate its result by multiplying the returned 1 with its own n, which gives 1. That's the next print, with an indentation of 2.
• And so on for all the higher recursion steps, from the inside to the outside. And that's how the second half of the V shape is formed.

The vital thing to understand is that the result values of the functions (and thereby the print) can only occur at the END of the complete recursion tree that comes below it. That's why the call print factorial 3 is in the first line, while its corresponding Returning 6 print is at the last line - after the whole inner tree is finished.

Hard to say if I picked up the crunchpoint you had trouble understanding with, but I hope this somehow helps.

EDIT: infinite recursion

As an answer to your question, here's an example that shows how each recursive function call will have its own variable scope and therefore not know about the variables set in an another instance of the same function, ending up in a RecursionError when the maxiumum recursion depth of the call stack is reached.

def foo():
    try:
        print("a = ", a)
        return   #  break out of loop
    except UnboundLocalError:
        print("no variable a. defining one now.")
        a = 3          
        foo()    #  try again

foo()

Answer 2

What might confuse you is that the recursive call does not happen at the end of the function as it is the case with your example f(). Which means, you have code that is executed before stepping into the recursion and code that is executed after stepping out of the recursion. Let's have a look at what happens in factorial:

The recursive step
Pick any recursion step i you like, which is not a base case. And then look at what the function does, without following through with the recursion.

For your factorial function, let's take the call factorial(i), where iis different from 0. That call does three things:

1. print 2*i spaces and "factorial i"
2. call factorial(i-1)
3. print 2*i spaces and "returning i"

So, the output is going to be:

(2*i spaces) factorial i
# everything factorial(i-1) outputs
(2*i spaces) returning i

You'll see that the output of factorial(i-1) gets sandwiched between the outputs of factorial(i). To perform the recursion, all you need to do is blow the pattern up. Let's take another step i-1 into account:

(2*i spaces) factorial i
(2*(i-1) spaces) factorial i-1
# everything factorial(i-1) outputs
(2*(i-1) spaces) returning i-1
(2*i spaces) returning i

So, you see that the indentation decreases with the calls since i decreases, which accounts for the v-shape.

The base case
It just prints

factorial 0
returning 1

Putting it all together
Now, you can find a verbal expression for what factorial(i) does: factorial(i) produces the v-shape output

      factorial i
    factorial i-1
  ...
factorial 0
returning 1
  ...
    returning i-1
      returning i

With it being true for the case i == 0, conclude from there that is is true for i+1 if you assume that is is correct for i using the list in the section recursive step. Done that, you can be sure that is how factorial(n) works for any n >= 0.