15B. Recursion

Recall that, whenever a function is called, it gets a new frame. So each call has its own variables. A consequence of that is that a function can call itself. It actually calls a copy of itself, with a new frame. A function that calls itself is said to be recursive, or to use recursion.

Example

---

Let's write a definition of function factorial(n) that returns n! = 1×2×…×n. The following two facts should be clear.

For example,

Those facts suggest the following definition of factorial, which works correctly as long as n ≥ 0.

  int factorial(int n)
  {
    if(n == 0)
    {
      return 1;
    }
    else
    {
      return n * factorial(n-1);
    }
  }

Detailed hand simulation of factorial

---

Let's use frames to simulate computation of factorial(3). To make the simulation easier to show, we break the statement in the else-part into statements. Here is the modified definition of factorial.

  int factorial(int n)
  {
    if(n == 0)
    {
      return 1;
    }
    else
    {
      int r = factorial(n-1);
      return n * r;
    }
  }

Since n is not 0, factorial moves to the else part and calls factorial(2).

The call to factorial(2) sees that n is 2, not 0, so it goes to the else part and calls factorial(1).

Since n is not 0, the call to factorial(1) calls factorial(0).

Since n is 0, the call to factorial(0) returns 1.

Factorial(1) returns 1*1 = 1.

Factorial(2) returns 2*1 = 2.

Finally, factorial(3) returns 3*2 = 6.

Exercises

---

1. What is the value of g(4), given the definition of g below? (Hint. Work out g(1), then g(2), then g(3), then g(4), in that order. Keep your work organized.)

     int g(int n)
     {
       if(n == 1) 
       {
         return 2;
       }
       else 
       {
         return g(n-1) + 3;
       }
     }
   

   Answer