f(5)
 5

f(4)   +   f(3)                f(2)+f(1)
  3     +   2                    1 + 1
f(3)+f(2)    f(1) + f(0)       f(1) + f(0)
  2 + 1        1  +  0          1   +  0
f(2)+f(1)
  1 +  1
f(1)+f(0)
  1 + 0


I computed f(5) once.
I computed f(4) once.
I computed f(3) twice.
I computed f(2) three times.


Let's say that I am computing fib(50).

fib(50) gets computed 1 time.
fib(49) " 1 time.
fib(48) " 2 times.
fib(47) " 3 times.
fib(46) " 5 times.
fib(47) " 8 times.


Computing fib(n) takes about fib(n) time.

Moral:  recursion is awesome, but it's not the best at everything.

We have a couple of options:

1.  Come up with a better algorithm.

2.  Use the same sucky algorithm, but add a hash table to it.

If you a hash table (or other data structure) to save results of
a computation, so that you never have to compute that value again,
this techinique is called "memoization."