$1.[14]$ From the encoding table as shown below:

Table $1$

a$)[3\mathrm{.}5]$ Build the prefix tree according to the Table $1$ above.

b$)[3\mathrm{.}5]$ Use Huffman code to assign new codewords to the symbols $($A$-$G$)$ above. Follow the convention that the group$/$symbol with higher probability is assigned a $\text{'}0\text{'}$ and that with lower probability is assigned a $\text{'}1\text{'}.$ List out all the codewords in Table $2.$

Table $2$

c$)[2]$ What is the average codeword length of the original coding $($Table $1)?$

d$)[2]$ What is the average codeword length by using Huffman code $($Table $2)?$

e$)[2]$ What is the minimum average number of bits needed to encode each symbol?

f$)[1]$ Using a fixed$-$length code, what will be the minimum average number of bits needed to encode each symbol?