int countLetter(String string, char character) {

if (string == null) {

throw new IllegalArgumentException("String must not be null");

}

if (string.length() == 0) {

return 0;

}

if (string.charAt(0) == character) {

return 1 + countLetter(string.substring(1), character);

} else {

return 0 + countLetter(string.substring(1), character);

}

}

System.out.println(countLetter("you are not alone", 'o'));

System.out.println(countLetter("you are not alone", 'e'));

This lesson introduces a new problem solving strategy called *recursion*.
Recursion represents an exciting new addition to our algorithm toolbox.
It can be hard to wrap your head around at first.
But we’ll go slow and, as always, use *lots* of examples.
What are we waiting for?

What is Recursion?

So far we’ve looked exclusively at iterative algorithms that solve a problem by repeating a series of steps.
The implementations of iterative algorithms are characterized by the use of looping constructs such as `for`

and `while`

.

Recursive algorithms work differently. They solve problems by breaking them down into smaller yet similar pieces, and the reassembling the solutions to the smaller problems into a solution to the larger problem. This will make more sense once we look at some examples!

But let’s start by examining both the general definition of recursion and how recursion is used in computer science.

Iteration v. Recursion

Learning to think recursively takes time and practice. You’ll get both!

But let’s start by comparing and contrasting an iterative and recursive solution to the same problem. That will both help us start to learn to think recursively, and identify the differences between the two approaches.

Is A Palindrome: Iterative

Let’s write an algorithm to determine if a `String`

is a palindrome, that is, whether it reads the same forwards and backwards.
First, let’s design an iterative algorithm:

// Iterative Palindrome

Is A Palindrome: Recursive

Next, let’s try and approach the problem *recursively*.

What does that mean?
A recursive solution tries to make the problem *smaller* in each step until it is trivial to solve.
Let’s think about how to apply that to palindrome detection.

// Recursive Palindrome

Compare and Contrast

To wrap up, let’s put the two approaches side by side and discuss the differences.

// Iterative Palindrome v. Recursive Palindrome

Recursive Strategies

A recursive implementation contains a function that calls *itself*.
At first, this may seem really weird.
But the trick is not to not let that throw you, and to just consider what happens.

Successful recursive algorithms *must* do three things correctly:

**Make the problem smaller.**If you don’t do this, the problem never gets small enough to solve!**Solve the smallest problem.**This is known as the base case.**Combine the results properly.**This is required to arrive at a correct solution.

Let’s examine another recursive implementation. We’ll identify the three requirements enumerated above, and trace its execution.

int countLetter(String string, char character) {

if (string == null) {

throw new IllegalArgumentException("String must not be null");

}

if (string.length() == 0) {

return 0;

}

if (string.charAt(0) == character) {

return 1 + countLetter(string.substring(1), character);

} else {

return 0 + countLetter(string.substring(1), character);

}

}

Factorial

The next homework problem has you complete a *recursive* implementation of factorial.
For reference, here is the *iterative* implementation that matches the homework problem:

public static long factorial(long input) {

// factorial gets big really quickly

if (input < 0 || input > 20) {

throw new IllegalArgumentException();

}

long result = 1;

for (long multiplier = 2; multiplier <= input; multiplier++) {

result *= multiplier;

}

return result;

}

assert factorial(0) == 1;

assert factorial(2) == 2;

assert factorial(4) == 24;

To complete the *recursive* implementation, consider the following questions:

- What is the smallest problem, or the base case? (You know that the factorial of 0 is 1).
- How do you make the problem smaller and combine the results? (You know that the factorial of n is n * n - 1).

Create a public class `Factorial`

that provides a single static method `factorial`

.
`factorial`

accepts a single `long`

and returns its factorial as a `long`

.
You can reject negative arguments and ones greater than 20 by throwing an `IllegalArgumentException`

.

You should submit a recursive solution. The factorial of 0 is 1, and this represents the base case. The factorial of n is n * the factorial of n - 1, and this represents the recursive step.

Create a class `RangeSum`

with a public static method `sum`

that accepts a single `int`

value and returns the sum
of all the integers in the range 1..value as an `int`

.
So, for example, given the input 10 you should return 55: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
You can reject arguments less than or equal to 0 and ones greater than 128 by throwing an
`IllegalArgumentException`

.

You should submit a *recursive* solution.
The range sum of 1 is 1, and this represents the base case.
The range sum of n is n + the range sum of n - 1, and this represents the recursive step.

Need more practice? Head over to the practice page.