1. Understanding Recursion

2. Introduction zRecursion is a powerful programming technique that provides elegant solutions to certain problems.

3. Introduction zRecursion is a powerful programming technique that provides elegant solutions to certain problems. zRecursion is a programming technique in which a method calls itself either directly, or indirectly through another method.

4. A Mathematical Example - Factorials zMathematical formulas often are expressed recursively.

5. A Mathematical Example - Factorials zMathematical formulas often are expressed recursively. zIn the following example, we will look in depth at factorials.

6. Definition of Factorial Factorials - ! The symbol for factorial is “!” - the exclamation mark. The factorial of a positive integer is the product of all nonnegative integers less than or equal to that number. Zero factorial is a special case and 0! = 1 From this definition, 5! is 120. 5! = 5. 4. 3. 2. 1 = 120 This formula often is defined recursively, for all nonnegative integers as: n! = n(n-1)! for n > 0; 0! = 1; Any number factorial is that number times the factorial of one less than that number.

7. A Closer Look Now, let’s look at the expression, n! = n * (n-1)! for n > 0; 0! = 1 You will notice that n! subtracts 1 from n, then recomputes the factorial of n-1. This is the recursion.

8. A Closer Look Now, let’s look at the expression, n! = n * (n-1)! for n > 0; 0! = 1 Also notice that the simplest case is 0! This is called the base case.

9. Base Cases zBase cases are important. A recursive method can solve only a base case.

10. Base Cases zBase cases are important. A recursive method can solve only a base case. zIf the method is called with a base case, it returns a result. If the methods is called with something other than the base case, the recursive method will decide what part it can accomplish, and then call itself to solve the rest of the problem.

11. Converting to Code To understand how to program recursively, we will convert the mathematical definition of factorial into code. n! = n · (n-1)! for n > 0; 0! = 1

12. Converting to Code To understand how to program recursively, we will convert the mathematical definition of factorial into code. We’ll start by creating a class, FactorialExample. public class FactorialExample { } n! = n · (n-1)! for n > 0; 0! = 1

13. Converting to Code For simplicity, we will add a main method. public class FactorialExample { } n! = n · (n-1)! for n > 0; 0! = 1

14. Converting to Code For simplicity, we will add a main method. The main method will create a FactorialExample object. public class FactorialExample { public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

15. Converting to Code We’ll add our recursive method, factorial. public class FactorialExample { public long factorial(long number) { } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

16. Converting to Code We now need to identify the base case; that is, the case the method factorial can solve without calling itself. public class FactorialExample { public long factorial(long number) { } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

17. Converting to Code In the formula above, we can use 0! = 1 as the base case. 0! is the simplest case. public class FactorialExample { public long factorial(long number) { } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

18. Converting to Code In the formula above, we can use 0! = 1 as the base case. 0! is the simplest case. public class FactorialExample { public long factorial(long number) { if (number == 0) return 1; } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

19. Converting to Code In the formula above, we can use 0! = 1 as the base case. 0! is the simplest case. However, 1! also equals 1. We can take advantage of this and change the code. public class FactorialExample { public long factorial(long number) { if (number == 0) return 1; } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

20. Converting to Code In the formula above, we can use 0! = 1 as the base case. 0! is the simplest case. However, 1! also = 1. We can take advantage of this and change the code. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

21. Converting to Code Now, we need to add recursion. We will look at the first part of the formula, n · (n-1)! If number is greater than 1, we need to compute n · (n-1)! public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

22. Converting to Code Now, we need to add recursion. We will look at the first part of the formula, n · (n-1)! If number is greater than 1, we need to compute n · (n-1)! public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } n! = n · (n-1)! for n > 0; 0! = 1

23. Examining the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } The best way to understand recursion is to step through the code.

24. Examining the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); } The best way to understand recursion is to step through the code. We will use 5! as our test case.

25. Examining the Code The best way to understand recursion is to step through the code. We will use 5! as our test case, and modify main slightly. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + “! = “ + answer); }

26. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The code starts by creating a FactorialExample object, fact.

27. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The testNumber variable is created and set to 5. The answer variable is created. answertestNumber 5-

28. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The factorial method is called. answertestNumber 5-

29. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The formal parameter number is created. answertestNumber 5- number 5

30. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The formal parameter number is not less than or equal to 1. answertestNumber 5- number 5

31. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } This line is the recursive call. answertestNumber 5- number 5 5

32. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } This line is the recursive call. The method will return the value of number (in this case, 5 ), multiplied by... answertestNumber 5- number 5 return: 5 *number 5

33. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } This line is the recursive call. The method will return the value of number (in this case, 5 ), multiplied by... The result of the method’s recursive call to itself. answertestNumber 5- number 5 return: 5 *number 5

34. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The factorial method is called, and another formal parameter number is created. This time the value of number is the previous formal parameter’s value ( number - 1 ) or 4. answertestNumber 5- number 5 return: 5 *number 5 4

35. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } The formal parameter number is not less than or equal to 1. answertestNumber 5- number 5 return: 5 *number 5 4

36. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } So, the method will return the value of number (in this case, 4 ), multiplied by... answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4

37. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } So, the method will return the value of number (in this case, 4 ), multiplied by... The result of the method’s recursive call to itself. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4

38. Stepping through the Code public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Another formal parameter number is created. This time the value of number is the previous formal parameter’s value (number - 1 ) or 3. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 3

39. Stepping through the Code The method returns 3 * the result of another recursive call. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); }

40. Stepping through the Code The method returns 3 * the result of another recursive call, with a new formal parameter. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } number 2

41. Stepping through the Code The method returns 2 * the result of another recursive call, answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); }

42. Stepping through the Code with a new formal parameter. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } number 1

43. Stepping through the Code The method finally can solve its base case. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } number 1

44. Stepping through the Code number is equal to 1. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } number 1

45. Stepping through the Code The method returns 1. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } number 1 return: 1

46. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code Control is returned to the calling method. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 1

47. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code The calling method now can return a value, in this case ( 2 * 1 ) or 2. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 return: 2 *number 2 1

48. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code Control is returned to the calling method. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 2

49. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code The calling method now can return a value, in this case ( 3 * 2 ) or 6. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 return: 3 *number 3 2

50. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code Control is returned to the calling method. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 6

51. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code The calling method now can return a value, in this case ( 4 * 6 ) or 24. answertestNumber 5- number 5 return: 5 *number 5 return: 4 *number 4 6

52. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code Control is returned to the calling method. answertestNumber 5- number 5 return: 5 *number 5 24

53. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code The last factorial method call will return control to the main method. The method will return the value of ( 5 * 24 ) or 120 answertestNumber 5- number 5 return: 5 *number 5 24

54. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code answer is assigned the value returned by the factorial method call, 120. answertestNumber 5120

55. public class FactorialExample { public long factorial(long number) { if (number <= 1) return 1; else return number * factorial(number - 1); } public static void main (String args[]) { FactorialExample fact = new FactorialExample(); long testNumber = 5; long answer; answer = fact.factorial(testNumber); System.out.println(testNumber + "! = " + answer); } Stepping through the Code The following is output to the screen: answertestNumber 5120 5! = 120

56. Summary zRecursion is a powerful programming technique that provides elegant solutions to certain problems. zRecursion is a technique in which a method calls itself either directly, or indirectly through another method. zBase cases are usually the simplest cases a recursive method can solve.

57. Summary zIf the method is called with a base case, it returns a result. If the methods is called with something other than the base case, the recursive method will decide what part it can accomplish, and then call itself to solve the rest of the problem. zThe best way to understand recursion is to step through the code.