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FIGURE 30. Diagram G, further expanded and with numbered nodes.
defined recursively by the pair of formulas
FIBO(n) = FIBO(n- 1) + FIBO(n-2) for n > 2
FIBO(l) = FIBO(2) = 1
Notice how new Fibonacci numbers are defined in terms of previous Fibonacci numbers.
We could represent this pair of formulas in an RTN (see Fig. 31).
FIGURE 31. An RTN for Fibonacci numbers.
Thus you can calculate FIBO(15) by a sequence of recursive calls on the procedure
defined by the RTN above. This recursive definition bottoms out when you hit FIBO(1)
or FIBO(2) (which are given explicitly) after you have worked your way backwards
through descending values of n. It is slightly awkward to work your way backwards,
when you could just as well work your way forwards, starting with FIBO(l) and FIBO(2)
and always adding the most recent two values, until you reach FIBO(15). That way you
don't need to keep track of a stack.
Now Diagram G has some even more surprising properties than this. Its entire
structure can be coded up in a single recursive definition, as follows:
Recursive Structures and Processes 144