A linear recursive function is a function that only makes a single call to itself each time the function runs (as opposed to one that would call itself multiple times during its execution). The factorial function is a good example of linear recursion.

;Scheme
            (define (factorial n)
            (if (= n 0)
            1
            (* n (factorial (- n 1)))
            )
            )          

//C++
            int factorial (int n)
            {
            if ( n == 0 )
            return 1;
            return n * factorial(n-1); // or factorial(n-1) * n
            }          

Tail recursion is a form of linear recursion. In tail recursion, the recursive call is the last thing the function does. Often, the value of the recursive call is returned. As such, tail recursive functions can often be easily implemented in an iterative manner; by taking out the recursive call and replacing it with a loop, the same effect can generally be achieved. In fact, a good compiler can recognize tail recursion and convert it to iteration in order to optimize the performance of the code.

A good example of a tail recursive function is a function to compute the GCD, or Greatest Common Denominator, of two numbers:

;Scheme
            (define (gcd m n)
            (cond
            ((< m n) (gcd n m))
            ((= (remainder m n) 0) n)
            (else (gcd n (remainder m n)))
            )
            )          

//C++
            int gcd(int m, int n)
            {
            int r;
            if (m < n) return gcd(n,m);
            r = m%n;
            if (r == 0) return(n);
            else return(gcd(n,r));
            }          

Some recursive functions don't just have one call to themself, they have two (or more). Functions with two recursive calls are referred to as binary recursive functions.

The mathematical combinations operation is a good example of a function that can quickly be implemented as a binary recursive function. The number of combinations, often represented as nCk where we are choosing n elements out of a set of k elements, can be implemented as follows:

;Scheme
            (define (choose n k)
            (cond
            ((= k 0) 1)
            ((= k n) 1)
            (else   (+     (choose (- n 1) k)  (choose (- n 1) (- k 1))   )     )
            )          

//C++
            int choose(int n, int k)
            {
            if (k == 0 || n == k) return(1);
            else return(choose(n-1,k) + choose(n-1,k-1));
            }					

An exponential recursive function is one that, if you were to draw out a representation of all the function calls, would have an exponential number of calls in relation to the size of the data set (exponential meaning if there were n elements, there would be O(aⁿ) function calls where a is a positive number).

A good example an exponentially recursive function is a function to compute all the permutations of a data set. Let's write a function to take an array of n integers and print out every permutation of it.

//C++
            void print_array(int arr[], int n)
            {
            int i;
            for(i=0; i<n; i++) printf("%d ", arr[i]);
            printf("\n");
            }
            void print_permutations(int arr[], int n, int i)
            {
            int j, swap;
            print_array(arr, n);
            for(j=i+1; j<n; j++) {
            swap = arr[i]; arr[i] = arr[j]; arr[j] = swap;
            print_permutations(arr, n, i+1);
            swap = arr[i]; arr[i] = arr[j]; arr[j] = swap;
            }
            } 			

In nested recursion, one of the arguments to the recursive function is the recursive function itself! These functions tend to grow extremely fast. A good example is the classic mathematical function, Ackermann's function. It grows very quickly (even for small values of x and y, Ackermann(x,y) is extremely large) and it cannot be computed with only definite iteration (a completely defined for() loop for example); it requires indefinite iteration (recursion, for example).

Try computing ackerman(4,2) by hand... have fun!

//C++
            int ackerman(int m, int n)
            {
            if (m == 0) return(n+1);
            else if (n == 0) return(ackerman(m-1,1));
            else return(ackerman(m-1,ackerman(m,n-1)));
            } 			

A recursive function doesn't necessarily need to call itself. Some recursive functions work in pairs or even larger groups. For example, function A calls function B which calls function C which in turn calls function A.

A simple example of mutual recursion is a set of function to determine whether an integer is even or odd. How do we know if a number is even? Well, we know 0 is even. And we also know that if a number n is even, then n - 1 must be odd. How do we know if a number is odd? It's not even!

//C++
            int is_even(unsigned int n)
            {
            if (n==0) return 1;
            else return(is_odd(n-1));
            }
            int is_odd(unsigned int n)
            {
            return (!is_even(n));
            } 			

Recursion is powerful! Of course, this is just an illustration. The above situation isn't the best example of when we'd want to use recursion instead of iteration or a closed form solution. A more efficient set of function to determine whether an integer is even or odd would be the following:

//C++
            int is_even(unsigned int n)
            {
            if (n % 2 == 0) return 1;
            else return 0;
            }
            int is_odd(unsigned int n)
            {
            if (n % 2 != 0) return 1;
            else return 0;
            }