1. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 1 Chapter 20 Recursion

2. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 2 Computing Factorial factorial(0) = 1; factorial(n) = n*factorial(n-1); n! = n * (n-1)! ComputeFactorial

3. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 3 Computing Factorial factorial(3) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

4. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 4 Computing Factorial factorial(3) = 3 * factorial(2) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

5. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 5 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

6. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 6 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

7. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 7 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

8. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 8 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

9. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 9 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) = 3 * 2 animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

10. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 10 Computing Factorial factorial(3) = 3 * factorial(2) = 3 * (2 * factorial(1)) = 3 * ( 2 * (1 * factorial(0))) = 3 * ( 2 * ( 1 * 1))) = 3 * ( 2 * 1) = 3 * 2 = 6 animation factorial(0) = 1; factorial(n) = n*factorial(n-1);

11. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 11 Trace Recursive factorial animation Executes factorial(4)

12. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 12 Trace Recursive factorial animation Executes factorial(3)

13. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 13 Trace Recursive factorial animation Executes factorial(2)

14. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 14 Trace Recursive factorial animation Executes factorial(1)

15. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 15 Trace Recursive factorial animation Executes factorial(0)

16. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 16 Trace Recursive factorial animation returns 1

17. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 17 Trace Recursive factorial animation returns factorial(0)

18. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 18 Trace Recursive factorial animation returns factorial(1)

19. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 19 Trace Recursive factorial animation returns factorial(2)

20. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 20 Trace Recursive factorial animation returns factorial(3)

21. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 21 Trace Recursive factorial animation returns factorial(4)

22. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 22 Towers of Hanoi F There are n disks labeled 1, 2, 3,..., n, and three towers labeled A, B, and C. F No disk can be on top of a smaller disk at any time. F All the disks are initially placed on tower A. F Only one disk can be moved at a time, and it must be the top disk on the tower.

23. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 23 Towers of Hanoi, cont.

24. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 24 Solution to Towers of Hanoi The Towers of Hanoi problem can be decomposed into three subproblems.

25. Liang, Introduction to Java Programming, Eighth Edition, (c) 2011 Pearson Education, Inc. All rights reserved. 0132130807 25 Solution to Towers of Hanoi F Move the first n - 1 disks from A to C with the assistance of tower B. F Move disk n from A to B. F Move n - 1 disks from C to B with the assistance of tower A. TowersOfHanoi