10.1 Description of Recursion

1. 10.1 Description of Recursion

2. Recursion • When a method contains a call to itself, this is referred to as recursion. • Consider the following class containing the skeleton of a static method: • public class MyClass • { • public static intmyMethod(intparameterOne) • { • intparameterTwo; • … • return myMethod(parameterTwo); • } • }

3. Recursion, cont. • myMethod() contains a call to itself. • Notice that syntactically, it’s not necessary to write the call as MyClass.mymethod(…) because the call occurs within code belonging to MyClass. • Recursion can be defined simply as a method that calls itself.

4. Rack of Pool Balls Example • The picture shown below represents a rack of pool balls. It will be used to illustrate how a recursive definition can be developed for the summation of positive integers. •  •  •  •  • 

5. Rack of Pool Balls Example, cont. • You can ask the following question as you move from the top row down: How does the total number of balls in the rack depend on the number of rows? • For a rack containing 5 rows, it’s easy enough to figure this out and put the information in a little table: • # of rows sum of balls in rows total number of balls in rack • 1 1 1 • 2 1 + 2 3 • 3 1 + 2 + 3 6 • 4 1 + 2 + 3 + 4 10 • 5 1 + 2 + 3 + 4 + 5 15 •  •  •  •  • 

6. The Math Part of Recursion • Using mathematical notation these sums can be shown as a function: • A different definition of the same function is given below. This definition contains two parts, a recursive case and a base case. • Recursive case: n > 1 • f(n) = n + f(n – 1) • Base case: n = 1 • f(1) = 1

7. The Math Part of Recursion, cont. • The idea is that for any integer n greater than 1, the function can be defined in terms of n and the same function applied to n – 1. • For n = 1, the value of the function is simply defined to be 1. • This is the base case, where the value of the function is no longer defined in terms of itself.

8. The Math Part of Recursion, cont. • Here is a definition of another mathematical function. • For x real and n some non-negative integer, x raised to the nth power is the product of n factors x: • xn = x * x * … * x • If you look in a math book, you may find the following recursive definition for this: • Recursive case: n > 0 • f(n) = x * f(n – 1) = x * xn-1 • Base case: n = 0 • f(0) = x0 = 1

9. The Math Part of Recursion, cont. • Lots of examples are possible. • In each you would observe that the definition of the function consists of two parts. • In the first, recursive part of the definition, the function is defined in terms of itself. • In the second part of the definition, the base case, the function is concretely defined for some fixed value. • In this second example the base case is defined for n = 0 rather than n = 1.

10. The Math Part of Recursion, cont. • In the examples above, n was a positive integer. • In the recursive part of the definition, f(n) was defined in terms of n and the f(n – 1). • In other words, f(n) was defined in terms of an application of the function to a value 1 less. • The base case defined f(n) for n = 0 or n = 1. • These examples illustrate the general structure of recursion. • n is decreased by 1 in each successive application of the recursive function. • It is critical that n reach the value of the base case so that the recursion will stop.