$3.(14$ points$)$ Suppose you want to encode messages containing only the following characters with their

given respective frequencies: B : $85,$ T : $50,$ P : $18,$ R : $20,$ C : $60$

$($a$)(2$ points$)$ What is the minimum length bit string required to encode each character with a distinct,

fixed$-$length code?

$($b$)(7$ points$)$ Construct the Huffman Tree for the characters with the given frequencies. $($Use the

convention that when merging two vertices, the vertex with the largest count goes on the left