$1.[$Optimality of Huffman Encoding$]$ Reconsider the proof of optimality for the Huffman encoding.

a$)$ You are given the following frequency table for the string "aaaaaaabbbccdddddeeeeeeeee":

a b c d e

$73259$

Is

a b c d e

$11001011011110$

an optimal prefix code? Explain why $($not$).$

b$)$ Construct the Huffman code H for the above frequency table.

How is each element encoded?

c$)$ Explain how to compute the total encoding length L$($H$)$ and the weighted external path length, respectively. Explain why they come out to the same result.

d$)$ Relook at the proof of optimality and observations $1$ and $2.$ Where do they appear in the proof of optimality?

e$)$ Explain why the proof of optimality works only for prefix codes.

f$)$ Explain what it means that the Huffman encoding is an optimal variable$-$length encoding. Does this mean that for any bitstring b$,$ we cannot find a shorter encoding than the Huffman encoding?