$(15$ points$)$ Huffman Encoding. You are given the frequencies of characters in a text

as A: $12,$ B: $9,$ C: $22,$ D: $30,$ E: $18,$ and F: $7.$

$($a$)(6$ points$)$ Construct a Huffman tree for encoding these characters.

$($b$)(2$ points$)$ Using the tree, encode the word "DECAF".

$($c$)(2$ points$)$ Decode the binary string $"1011001110"$

$($d$)(5$ points$)$ Suppose that, in addition to frequency, the encoding should prioritize vowels

$($A$,$ E$,$ I, O$,$ U$),$ giving vowels shorter codes regardless of their frequency. Describe

how you would modify the Huffman tree construction process to incorporate this new

prioritization. What should change in the encoding algorithm procedure?