2. (15 pts) Alice is writing a secret message to Bob and wants to prevent it from being understood by eavesdroppers. She decides to use Huffman encoding to encode the message, based on the symbol frequencies that are given by the Lucas numbers. The nth Lucas number is defined as Ln-Ln-1 + Ln-2 for n > 1 with base cases Lo = 2 and L1 1 (a) For an alphabet of a, b, c, d, e, f.g,h, i with frequencies given by the first E Lucas numbers, give an optimal Huffman code and the corresponding encoding tree for Alice to use. (b) Generalize your answer to (2a) and give the structure of an optimal code when the frequencies are the first n Lucas numbers