Huffman Encoding

Consider a message from an alphabet containing six symbols: $=\{A,B,C,D,E,F\}.$ These symbols each have the following relative frequencies,

$\frac{?}{\text{b}\text{a}\text{r}}(f)(A)=\mathrm{.}25\frac{?}{b}$

$ar(f)(B)=\mathrm{.}10\frac{?}{b}$

$ar(f)(C)=\mathrm{.}15\frac{?}{b}$

$ar(f)(D)=\mathrm{.}05\frac{?}{b}$

$ar(f)(E)=\mathrm{.}10\frac{?}{b}$

$ar(f)(F)=\mathrm{.}35$

a$.(5$ pts$)$ What is the minimum number of bits required per symbol to construct a fixed$-$length encoding over this alphabet? If you transmit a message

containing $1,000,000$ symbols using this encoding, how many bits do you need to encode the message?

b$.(10$ pts$)$ Construct a Huffman Encoding over these symbols for a message of $1,000,000$ symbols. Show each step of the process, including the partial

Huffman Trees.

c$.(5$ pts$)$ How many bits are needed to transmit the message with $1,000,000$ symbols using your Huffman encoding? Additionally, suppose that for some

reason the frequencies of symbols were estimated in error, and it's actually the case that $\frac{?}{\text{b}\text{a}\text{r}}(f)(D)=\mathrm{.}35$ and $\frac{?}{\text{b}\text{a}\text{r}}(f)(F)=\mathrm{.}05.$ How many bits would be

required in this case, using the Huffman Encoding you determined in part b$.$ In this case, would the fixed$-$length encoding be better?