Class Notes Week 11


Monday Wednesday Friday


Any function that sometimes calls itself is known as a recursive function. In most cases, recursion can be considered an alternative to iteration (loops). Instead of defining a loop structure, recursion defines a problem as a smaller problem until it reaches an end point. There are three requirements for a successful recursive function:

  1. Base case: each function must have one or more base cases where the function does not make a recursive call to itself. Instead, the problem has reached its smallest point, and we can begin returning back the small solutions.

  2. Recursive case: the function makes one (or more) calls to itself. The call must use a different argument to the function (or else we keep making the same recursive call over and over again!). Typically, the problem gets 1 smaller.

  3. All possible recursive calls must be guaranteed to hit a base case. If you miss one, your program is susceptible to an infinite recursion bug!

Example: print a message

$ cd ~/cs21/inclass/w11-recursion/

The program prints a given message n times. We have already seen how to solve this problem with a loop, but let's explore how we can solve it without a loop using recursion. To understand what is happening with recursion, it is helpful to draw a stack diagram. The key concept is that Python treats the recursive call the same as any function call - it places a new stack frame on top and begins executing the new function. The only difference is the function in this case has the same name.

Example: summing integers

$ cd ~/cs21/inclass/w11-recursion/

We are going to write some iterative and recursive versions of the same function. Iterative versions use loops, recursive versions do not use loops, and instead contain calls to themselves but on a smaller problem. I have given you the code for an algorithm we have seen before - a for loop that sums all integers between 1 and n:

def iterativeSum(n):
    total = 0
    for i in range(n):
      total =  i+1
    return total

Together, we will write a recursive function that accomplishes the same task.

Exercise: factorial

Recall the iterative version (using a loop) of factorial that takes a positive int value, n, and returns the product of the first n integers. Try writing a recursive version of the factorial function.


Before moving on to more recursion examples, we will introduce a new python operator called slicing that can be particularly useful when working with recursion. Slicing lets you easily look at slices, of strings or lists. The general syntax is mystr[i:j]. This gives the substring of mystr starting at index i and running up to but not including index j.

$ python3
>>> hw = "hello world"
>>> hw[1]
>>> hw[7:10]
>>> hw[0:5]
>>> hw[6:]
>>> hw[:5]

You can also omit i or j. mystr[i:] gives you the substring starting at index i and running to the end of that string. mystr[:j] gives you the substring up to but not including the j-th character.

Note: slicing can also be used with lists e.g. if ls = [1,2,3,4,5], then ls[2:4] == [3,4].


A palindrome is a word or phrase that appears the same when read backwards or forwards. Examples include: "kayak", "taco cat", "Top spot", "Now I see bees. I won!", and "may a moody baby doom a yam". In, write a recursive function isPalindrome that takes a string as input and returns True if the string is a valid palindrome, or false otherwise. Think first about base cases. When is your program free to not recurse? Then think about the general case. When do you know a string is not a palindrome?

Exercise: interpret a recursive function

Open Explain the program to your neighbor. What is the recursive function doing? Can you identify the base vs recursive case? Are you sure that all recursive calls eventually hit the base case?

Tips on recursion

For any recursive problem, think of these questions:

  1. What is the base case? In what instances is the problem small enough that we can solve the problem directly? Sometimes you may need more than one base case, but rarely do you need several.
  2. What is the recursive call? Make one or calls in the function to itself. It is very important to use different arguments (usually "smaller") in the recursive call otherwise you obtain infinite recursion.
  3. Do all possible chains of recursion lead to a base case?
  4. If a recursive function is supposed to return a value, make sure all cases return a value and that recursive calls do something with the return value of the smaller problem before returning.