$(10$ points$)$ Understanding the Merge Sorting algorithm. Read Example $1$

on page $122-123$ of our textbook Applied Combinatorics, $6$th edition by Alan

Tucker, which explains how to do the merge$-$sort algorithm. Based on your

understanding of that example, answer the following questions.

$($a$)$ Apply the merge sorting algorithm to the list:

$L=(15,27,4,-7,9,13,8,28,12,20,-80)$

$($b$)$ Assume two sub$-$lists have lengths which add up to $n.$ Show that it takes

at most $(n-1)$ comparisons to correctly merge the two lists by describing

a procedure which uses at most that many.

$($c$)$ Obtain a lower bound on the height of the merge$-$sort tree for any list $L$

with length $n.$ Combine this bound with your answer to $B$ to show that

the merge$-$sort algorithm completes in approximately $nlo{g}_2(n)$ time.