11.3.7 Recursion without Deferments

Let’s consider again what happens with the triangle-recursively function. We will find that the intermediate calculations are deferred until all can be done.

Here is the function definition:

(defun triangle-recursively (number)
  "Return the sum of the numbers 1 through NUMBER inclusive.
Uses recursion."
  (if (= number 1)                    ; do-again-test
      1                               ; then-part
    (+ number                         ; else-part
       (triangle-recursively          ; recursive call
        (1- number)))))               ; next-step-expression

What happens when we call this function with an argument of 7?

The first instance of the triangle-recursively function adds the number 7 to the value returned by a second instance of triangle-recursively, an instance that has been passed an argument of 6. That is to say, the first calculation is:

(+ 7 (triangle-recursively 6))

The first instance of triangle-recursively—you may want to think of it as a little robot—cannot complete its job. It must hand off the calculation for (triangle-recursively 6) to a second instance of the program, to a second robot. This second individual is completely different from the first one; it is, in the jargon, a “different instantiation”. Or, put another way, it is a different robot. It is the same model as the first; it calculates triangle numbers recursively; but it has a different serial number.

And what does (triangle-recursively 6) return? It returns the number 6 added to the value returned by evaluating triangle-recursively with an argument of 5. Using the robot metaphor, it asks yet another robot to help it.

Now the total is:

(+ 7 6 (triangle-recursively 5))

And what happens next?

(+ 7 6 5 (triangle-recursively 4))

Each time triangle-recursively is called, except for the last time, it creates another instance of the program—another robot—and asks it to make a calculation.

Eventually, the full addition is set up and performed:

(+ 7 6 5 4 3 2 1)

This design for the function defers the calculation of the first step until the second can be done, and defers that until the third can be done, and so on. Each deferment means the computer must remember what is being waited on. This is not a problem when there are only a few steps, as in this example. But it can be a problem when there are more steps.