Q$4.$ Assume that the numbers given below represent counts of letters in the hundreds from a file.

For example, in the file there will be exactly $20*100$ occurrences of the letter $$a$,11*100$

occurrence of the letter $$c$,$ etc. $(25$ points$)$

a: $20,$ c:$11,$ d:$2,$ e: $10,$ o:$15,$ m:$8,$ s:$10,$ t:$22,$ u: $2$

a$)$ What is an optimal Huffman code based on the following set of frequencies?

$1)$ Draw the tree. Show your work at every step.

$2)$ Fill in the table on the right the Huffman encoding for each letter.

$3)$ We encode the file using the Huffman codes produced above. How much memory will the file require with this encoding?

b$)$ Fixed$-$length encoding:

$1)$ Fill in the table the fixed$-$length encoding for each letter.

$2)$ What will be the file size when the fixed$-$length encoding is used?