Week 10, Monday: Recursion


Recursion is when a function calls another copy of itself to finish the work. Some algorithms, like the merge sort (or any divide-and-conquer algorithms), are more easily expressed using recursion. This week we will practice writing recursive functions.

Obviously, if the function calls another copy of itself, the other copy may call yet another copy, and so on. This leads to repetition, very similar to a loop. In fact, designed correctly, a recursive function can simply replace a for loop. So all of the functions we have written so far (sum a list, blastoff, find the max of a list, etc) can be written using recursion (and not using a for loop or a while loop).

Here is the simple blastoff(n) function, to show a countdown from some number n, using a for loop:

def blastoff(n):
  for i in range(n,0,-1):

Thinking recursively, you may notice that blastoff(10) is really just this:


And blastoff(9) is this:


And so on. The key to writing an algorithm recursively: handle the first item or case, then let recursion handle the rest. For the above, given an integer n, we would:

print(n)           # take care of n
blastoff(n-1)      # let recursion handle the rest

Also note, each subsequent call to blastoff() has a smaller value of n. Eventually we will call blastoff(2), then blastoff(1), and then, finally, blastoff(0). At that point, we want to just print "blastoff" and stop the recursion. This is formally called the base case, and is usually written with an if/else clause.

Here is the full recursive blastoff function:

def recursiveblastoff(n):
  if n <= 0:

Notice the base case stops the recursion (does not call another copy). Also notice the non-base case (the else clause) heads toward the base case by calling another copy, but with n-1 as the argument.

Sometimes, to understand recursion, it helps to think about the stack frames. If the above function were called with n=4, how many frames would be put on the stack? Try it in the python tutor! Here is a screenshot of the stack frames, when the recursion has reached the base case:

recursiveblastoff stack frames

Here is another example, this one from mathematics: the factorial(n) function is defined as n*n-1*n-2*...*3*2*1. So, for example, factorial(5) is just 5*4*3*2*1. Notice, however, that factorial(5) can be written as just 5*factorial(4), since factorial(4) = 4*3*2*1. If factorial(0) = 1, can you write a recursive factorial function? You can test your function against the factorial function in the math library:

>>> from math import factorial
>>> factorial(5)
>>> factorial(8)

Note: both the factorial() and the blastoff() functions can easily be written without using recursion. We are just using them as simple examples. This week we will try to write as many recursive functions as we can, just to learn recursion and get used to thinking recursively. Later in the week we will look at some algorithms that are more easily expressed using recursion.

thinking recursively

I think the hardest part about writing recursive functions is the first step, where you need to express the problem in a recursive fashion. Try to keep the following in mind: handle the first case, then let recursion handle the rest. Once you have that part figured out, it is usually easy to write the base case. Let's try a few examples:

sum a list

Given a list of numbers, L, write a recursive function (no for loops!) to sum the list and return the result.

The first step is usually the hardest...the sum of a list is just L[0] + the sum of the rest of the list. Using python slicing, it is easy to grab the rest of the list: L[1:]. If sum(L) calls sum(L[1:]), what is the base case we are heading toward? And if we reach the base case, what should the function return?

flipping coins

Given a positive integer n, flip a coin n times and return the number of heads flipped. Write a recursive coinflip(n) function (again, no for loops!).

Note that n flips is just one coinflip plus n-1 more flips. So our function, in the non-base case, needs to flip 1 coin, figure out if it was heads or tails, then let recursion flip the other n-1 times, add those to the result of the first flip, and return the full result.

Each subsequent call to coinflip() has one lower value of n, so what is the base case?

counting items in a list

Given a list of numbers, L, and a value, x, write a recursive function to count how many of x are in L. For example, calling count(range(10), 99) would return 0, and count([1,2,8,2,2], 2) would return 3.

Again, handle the first item in the list, L[0], then let recursion handle the rest of the list (L[1:]).