Understanding Recursion

1
Understanding Recursion
2
Introduction

• Recursion is a powerful programming technique
  that provides elegant solutions to certain
  problems.

3
Introduction

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

4
A Mathematical Example -Factorials

• Mathematical formulas often are expressed
  recursively.

5
A Mathematical Example -Factorials

• Mathematical formulas often are expressed
  recursively.
• In 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 gt 0 0! 1
• Any number factorial is that number times the
  factorial of one less than that number.

7
A Closer Look

• Now, lets look at the expression,
• n! n (n-1)! for n gt 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, lets look at the expression,
• n! n (n-1)! for n gt 0 0! 1
• Also notice that the simplest case is 0! This is
  called the base case.

9
Base Cases

• Base cases are important. A recursive method can
  solve only a base case.

10
Base Cases

• Base cases are important. A recursive method can
  solve only a base case.
• If 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
n! n (n-1)! for n gt 0 0! 1

• To understand how to program recursively, we will
  convert the mathematical definition of factorial
  into code.

12
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample

• To understand how to program recursively, we will
  convert the mathematical definition of factorial
  into code.
• Well start by creating a class, FactorialExample.

13
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample

• For simplicity, we will add a main method.

14
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public
static void main (String args)
FactorialExample fact new
FactorialExample()

• For simplicity, we will add a main method.
• The main method will create a FactorialExample
  object.

15
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) public static void
main (String args) FactorialExample fact
new FactorialExample()

• Well add our recursive method, factorial.

16
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) public static void
main (String args) FactorialExample fact
new FactorialExample()

• We now need to identify the base case that is,
  the case the method factorial can solve without
  calling itself.

17
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) public static void
main (String args) FactorialExample fact
new FactorialExample()

• In the formula above, we can use 0! 1 as the
  base case. 0! is the simplest case.

18
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) if (number
0) return 1 public static void main
(String args) FactorialExample fact
new FactorialExample()

• In the formula above, we can use 0! 1 as the
  base case. 0! is the simplest case.

19
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) if (number
0) return 1 public static void main
(String args) FactorialExample fact
new FactorialExample()

• 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.

20
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) if (number lt
1) return 1 public static void main
(String args) FactorialExample fact
new FactorialExample()

• 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.

21
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) if (number lt
1) return 1 public static void main
(String args) FactorialExample fact
new FactorialExample()

• 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)!

22
Converting to Code
n! n (n-1)! for n gt 0 0! 1
public class FactorialExample public long
factorial(long number) if (number lt
1) return 1 else return number
factorial(number - 1) public static void main
(String args) FactorialExample fact
new FactorialExample()

• 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)!

23
Examining the Code
The best way to understand recursion is to step
through the code.
public class FactorialExample public long
factorial(long number) if (number lt
1) return 1 else return number
factorial(number - 1) public static void main
(String args) FactorialExample fact
new FactorialExample()
24
Examining the Code
The best way to understand recursion is to step
through the code.
public class FactorialExample public long
factorial(long number) if (number lt
1) return 1 else return number
factorial(number - 1) public static void main
(String args) FactorialExample fact
new FactorialExample()
We will use 5! as our test case.
25
Examining the Code
The best way to understand recursion is to step
through the code.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber !
answer)
We will use 5! as our test case, and modify main
slightly.
26
Stepping through the Code
The code starts by creating a FactorialExample
object, fact.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
27
Stepping through the Code
The testNumber variable is created and set to 5.
The answer variable is created.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
answer
testNumber
28
Stepping through the Code
The factorial method is called.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
29
Stepping through the Code
The formal parameter number is created.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
30
Stepping through the Code
The formal parameter number is not less than or
equal to 1.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
31
Stepping through the Code
This line is the recursive call.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
32
Stepping through the Code
This line is the recursive call. The method will
return the value of number (in this case, 5),
multiplied by . . .
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
33
Stepping through the Code
This line is the recursive call. The method will
return the value of number (in this case, 5),
multiplied by . . . The result of the methods
recursive call to itself.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
34
Stepping through the Code
The factorial method is called, and another
formal parameter number is created. This time
the value of number is the previous formal
parameters value (number - 1) or 4.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
35
Stepping through the Code
The formal parameter number is not less than or
equal to 1.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
36
Stepping through the Code
So, the method will return the value of number
(in this case, 4), multiplied by . . .
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
37
Stepping through the Code
So, the method will return the value of number
(in this case, 4), multiplied by . . . The result
of the methods recursive call to itself.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
38
Stepping through the Code
Another formal parameter number is created.
This time the value of number is the previous
formal parameters value (number - 1) or 3.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
39
Stepping through the Code
The method returns 3 the result of another
recursive call.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
40
Stepping through the Code
The method returns 3 the result of another
recursive call, with a new formal parameter.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
41
Stepping through the Code
The method returns 2 the result of another
recursive call,
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
42
Stepping through the Code
with a new formal parameter.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 2
return 4
43
Stepping through the Code
The method finally can solve its base case.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 2
return 4
44
Stepping through the Code
number is equal to 1.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 2
return 4
45
Stepping through the Code
The method returns 1.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 1
return 2
return 4
46
Stepping through the Code
Control is returned to the calling method.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 2
1
return 4
47
Stepping through the Code
The calling method now can return a value, in
this case ( 2 1 ) or 2.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 2
1
return 4
48
Stepping through the Code
Control is returned to the calling method.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
2
return 4
49
Stepping through the Code
The calling method now can return a value, in
this case ( 3 2 ) or 6.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
2
return 4
50
Stepping through the Code
Control is returned to the calling method.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
6
51
Stepping through the Code
The calling method now can return a value, in
this case ( 4 6 ) or 24.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
return 4
6
52
Stepping through the Code
Control is returned to the calling method.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
24
53
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
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
24
54
Stepping through the Code
answer is assigned the value returned by the
factorial method call, 120.
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
answer
testNumber
120
55
Stepping through the Code
The following is output to the screen
public class FactorialExample public long
factorial(long number) if (number lt
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(testNumb
er) System.out.println(testNumber "! "
answer)
5! 120
answer
testNumber
120
56
Summary

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

57
Summary

• If 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.
• The best way to understand recursion is to step
  through the code.