1. Slides prepared by Rose Williams, Binghamton University Chapter 11 Recursion

2. © 2006 Pearson Addison-Wesley. All rights reserved11-2 Recursion Versus Iteration Recursion is not absolutely necessary –Any task that can be done using recursion can also be done in a nonrecursive manner –A nonrecursive version of a method is called an iterative version An iteratively written method will typically use loops of some sort in place of recursion A recursively written method can be simpler, but will usually run slower and use more storage than an equivalent iterative version

3. © 2006 Pearson Addison-Wesley. All rights reserved11-3 Iterative version of writeVertical

4. © 2006 Pearson Addison-Wesley. All rights reserved11-4 Recursive Methods that Return a Value Recursion is not limited to void methods A recursive method can return a value of any type An outline for a successful recursive method that returns a value is as follows: –One or more cases in which the value returned is computed in terms of calls to the same method –the arguments for the recursive calls should be intuitively "smaller" –One or more cases in which the value returned is computed without the use of any recursive calls (the base or stopping cases)

5. © 2006 Pearson Addison-Wesley. All rights reserved11-5 Another Powers Method The method pow from the Math class computes powers –It takes two arguments of type double and returns a value of type double The recursive method power takes two arguments of type int and returns a value of type int –The definition of power is based on the following formula: x n is equal to x n-1 * x

6. © 2006 Pearson Addison-Wesley. All rights reserved11-6 Another Powers Method In terms of Java, the value returned by power(x, n) for n>0 should be the same as power(x, n-1) * x When n=0, then power(x, n) should return 1 –This is the stopping case

7. © 2006 Pearson Addison-Wesley. All rights reserved11-7 The Recursive Method power (Part 1 of 2)

8. © 2006 Pearson Addison-Wesley. All rights reserved11-8 The Recursive Method power (Part 1 of 2)

9. © 2006 Pearson Addison-Wesley. All rights reserved11-9 Evaluating the Recursive Method Call power(2,3)

10. © 2006 Pearson Addison-Wesley. All rights reserved11-10 Example 1: Factorial Let us construct a recursive version of a program that evaluates the factorial of an integer, i.e. fact(n) = n! fact(0) is defined to be 1. i.e. fact(0) = 1. This is the base step. For all n > 0, fact(n) = n * fact(n – 1). This is the recursive step. As an example, consider the following: fact(4)= 4 * fact(3) = 4 * (3 * fact(2)) = 4 * (3 * (2 * fact(1))) = 4 * (3 * (2 * (1 * fact(0)))) = 4 * (3 * (2 * (1 * 1))) = 4 * (3 * (2 * 1)) = 4 * (3 * 2) = 4 * 6 = 24 Not in the BOOK 

11. © 2006 Pearson Addison-Wesley. All rights reserved11-11 Example 1: Program import java.io.*; public class Factorial { public static void main(String[] args) throws IOException { BufferedReader in = new BufferedReader(new InputStreamReader(System.in)); System.out.println("Enter an integer: "); long fact = ((long) Integer.parseInt(in.readLine())); long answer = factorial(fact); System.out.println("The factorial is: " + answer); } public static long factorial(long number) { if(number == 0) //base step return 1; else//recursive step return number * factorial(number - 1); } Not in the BOOK 

12. © 2006 Pearson Addison-Wesley. All rights reserved11-12 Example 2: Fibonacci Numbers Consider the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13 Except for the first two integers, each integer is the sum of the previous two integers. This sequence is known as fibonacci (pronounced fibo – naachee) numbers. Fibonacci numbers have important applications in computer science and mathematics. A recursive solution for calculating the n th (n > 1) is as follows: BASE CASE:fib(n) = 1 if n = 1 or 2 (n < 2) RECURSIVE CASE:fib(n) = fib(n – 1) + fib(n – 2) Not in the BOOK 

13. © 2006 Pearson Addison-Wesley. All rights reserved11-13 Example 2: Program import java.io.*; public class Fibonacci { public static void main(String[] args) throws IOException { BufferedReader in = new BufferedReader(new InputStreamReader(System.in)); System.out.println("Enter an integer: "); int n = Integer.parseInt(in.readLine()); int answer = fib(n); System.out.println("The " + n + "th fibonacci number is: " + answer); } public static int fib(int number) { if(number <= 2) //base step return 1; else//recursive step return fib(number - 1) + fib(number - 2); } Not in the BOOK 