Exercise 9, page 205. just write out the recursive

1. (3 points) Exercise 9, page 205. Just write out the recursive method gcd; you aren’t
required to write or run a test program. 
2. (3 points) Exercise 17, page 209. 
3. (4 points) Draw a recursion tree for the call binarySearch(5, 0, 8). Use the
recursive version of the method found on p. 174 of your textbook.
Label each node of the recursion tree with the indices of the subarray being searched
(given by the parameters first and last). Assume the array values contains the

[-2 1 3 5 7 12 18 20 23]

4. (2 points) How many total calls are made to binarySearch? What is the depth of
recursion for this call? Here depth of recursion = longest path in the recursion tree.
5. (4 points) In class we saw two ways to formulate the choose function, one with two
recursive calls, and another needing only one recursive calls. Write both versions (call
them choose1 and choose2, for the number of recursive calls) as static methods in a
class along with a main method for testing. For both, be sure to check the validity of the
arguments and throw IllegalArgumentException if invalid. Your test will prompt
the user for n and m and return the results of running both versions of choose.  

Record the results of testing your code with arguments (4,2) and (6,4). Then test them for
(15,8) and (35,20).  On some computers the last call may take several minutes.  If any of
your answers are negative, explain why.

6. (2 points) Java provides the primitive type long for integral computations which cannot
be expressed using int.  Long integers hold values from more than 92 quintillion to less
than -92 quintillion.  Rewrite each choose method so that it returns value of type long. 
Rerun each computation from the previous problem and record the results.

7. (2 points) How many total calls are made to choose2(6,4)? What is the depth of
recursion for this call? Use a recursion tree to answer this question (extra credit for
turning in a drawing of the tree).

8. (4 points) For each of the algorithms gcd, contains, binarySearch,
and choose2, say if they are tail recursive or not.

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