Recursion: functions that call themselves#

Recursion is when things refer back to themselves; we’ll be studying recursive functions. That is, we’re going to learn about functions that call themselves.

Here’s a simple recursive function that counts up to ten:

How does this function work? Running it in PythonTutor is a helpful visualization. Each time countup is called, it checks to see if it’s hit its limit, n > 10. If it has, it just returns. If it’s not yet at the limit, countup prints out n and calls itself with n one greater.

Here’s another example, where we’ll count down to 0, summing numbers as we go.

Again, PythonTutor is a great help. Here’s what happens:

sumup(10) — since 10 > 0, we go to the recursive case, computing…
10 + sumup(9) — since 9 > 0, we go to the recursive case, computing…
10 + 9 + sumup(8) — since 8 > 0, we go to the recursive case, computing…
10 + 9 + 8 + sumup(7) — since 7 > 0, we go to the recursive case, computing…
…and so on, until we have…
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + sumup(1) — since 1 > 0, we go to the recursive case, computing…
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + sumup(0) — and it’s not the case that 0 > 0, so sumup(0) returns 0, and we have…
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 which is 55.

In recursive functions, we say something is a base case if it doesn’t make a recursive call, and we say it’s a recursive case otherwise. In the case of countup, if n > 10: return is a base case. For sumup, the base case is if n <= 0: return 0.