Exercise 5.29: Monitor the stack operations in
the tree-recursive Fibonacci computation:


(define (fib n)
  (if (< n 2)
      n
      (+ (fib (- n 1)) (fib (- n 2)))))


 Give a formula in terms of 
  n
 for the maximum depth of the stack required to
compute 
  
    Fib
    (
    n
    )
  
 for 
  
    n
    ≥
    2
  
.  Hint: In 1.2.2 we
argued that the space used by this process grows linearly with 
  n
.

 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
.