Monitor the stack operations in the tree-recursive Fibonacci computation:
(define (fib n) (if (< n 2) n (+ (fib (- n 1)) (fib (- n 2)))))
a. Give a formula in terms of n for the maximum depth of the stack required to compute Fib(n) for n ≥ 2. Hint: In section 1.2.2 we argued that the space used by this process grows linearly with n.
b. Give a formula for the total number of pushes used to compute Fib(n) for n ≥ 2. You should find that the number of pushes (which correlates well with the time used) grows exponentially with n. Hint: Let S(n) be the number of pushes used in computing Fib(n). You should be able to argue that there is a formula that expresses S(n) in terms of S(n - 1), S(n - 2), and some fixed ''overhead'' constant k that is independent of n. Give the formula, and say what k is. Then show that S(n) can be expressed as a Fib(n + 1) + b and give the values of a and b.
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