UNIVERSITY AT BUFFALO, THE STATE UNIVERSITY OF NEW YORK
The Department of Computer Science & Engineering

STUART C. SHAPIRO: CSE 115 C

To write a recursive method for a problem takes two steps:

1. Base Case

The base case is an instance of the problem that doesn't require recursion.

2. Recursive Step

The recursive step solves the instance of the problem by some computation based on calling the method itslef with an instance of the problem that is closer to the base case.

Example: the factorial of n, abbreviated n!, is the product of all positive integers from 1 to n, or stated recursively, 1! = 1 and n! = n*(n-1)!
By common definition, 0!=1, and we can define n! to also be 1 if n is negative.
Factorial in Java:

public static int fact(int n) {
    if (n<= 0) {return 1;}        // Base case
    else {return n * fact(n-1);}  // Recursive step

To make sure that the recursive method is correct requires three steps:

1. The base case must be correct.
2. The recursive call must be of an instance of the problem closer to the base case.
3. The recursive step must be correct assuming that the recursive call yields the correct answer.

1. The base case is correct, because if n<=0, n!=1 by definition.
2. The recursive call is of an instance of the problem closer to the base case, because if n>0, n-1 is closer to 0 than n is.
3. The recursive step is correct assuming that the recursive call yields the correct answer, because of fact(n-1) = 1*...*(n-1), then n*fact(n) = 1*...*(n-1)*n = 1*...*n.