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Computer Science 211
Data Structures
Mount Holyoke College
Fall 2009
Topic Notes: Recursion and Mathematical Induction
Recursion
An important tool when trying to solve a problem is the ability to break the problem down into
some number of smaller sub-problems, the solutions to which you can use to solve the original
problem.
Oftentimes, those sub-problems look a lot like the original problem. In fact, they might be the
same
problem, just on a smaller set of input data.
These kinds of problems often have a
self-referential
or
recursive
solution.
It’s a strange idea at first – calling the method you’re writing before you’re done writing it. Well,
if my method needs to call my method to finish up, how am I ever going to get anywhere?
Many algorithms may be recursive. Once you are used to them, they can be easier to understand
(and prove correct) than iterative algorithms.
Even if you’re fairly comfortable with recursion, I expect that few of you have formally proven
properties about recursive algorithms before. We will be doing some of that soon.
We start with a simple and classically recursive example: computing a factorial.
n
! =
n
·
(
n
−
1)
·
...
·
1
We could write a simple method to compute this with a for or while loop. But it is just begging to
be solved recursively.
n
!
is nothing more than
n
·
(
n
−
1)!
. So to compute it, all we do is compute
(
n
−
1)!
(which is
certainly easier than computing
n
!
) and then multiply by
n
and we have the answer.
public static int factorial(int n) {
return n
*
factorial(n-1);
}
That’s great. We were writing a method to compute factorials anyway, so why not call it? Assum-
ing we know how to compute
(
n
−
1)!
, we now can compute
n
!
.
Problem: what about 2?
2! = 2
∗
(1!) = 2
∗
1
∗
(0!) = 2
∗
1
∗
0
∗
(
−
1!)
...
That won’t work. We
need to stop the recursion somehow.
We need a
base case
. Well, 1!=1, so let’s stop our factorial when it gets to 1:

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Data Structures
Fall 2009
public static int factorial(int n) {
if (n == 1) return 1;
return n
*
factorial(n-1);
}
The keys to a successful recursive solution: identify the base case and make sure the recursive step
is making
progress
toward the solution (closer to the base case).
Pros and Cons of Recursion

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