Final answer:
The maximal length of a Huffman encoded word for an alphabet of n symbols is n - 1. This occurs in a scenario where each subsequent symbol has a progressively lower frequency, resulting in a chain-like Huffman tree. However, Huffman encoding generally aims for the shortest overall message length, making this scenario unlikely.
Step-by-step explanation:
The maximal length of a code word possible in a Huffman encoding of an alphabet of n symbols is n - 1. Huffman encoding is a method for creating a binary tree where each symbol of the alphabet is represented by a unique binary sequence. The lengths of the code words are determined by the frequencies of the symbols: the more frequent a symbol, the shorter the code word.In the least efficient scenario, you would build a tree where every new symbol is added as a leaf node to the rightmost node, causing a chain-like structure. This occurs when each symbol has less frequency than the sum of all previously considered symbols together. In this case, the final symbol to be encoded (the nth symbol) would be at the bottom of this chain, making its length n - 1 bits. This scenario provides the longest possible Huffman code word for a given alphabet size.
However, it is important to note that in most practical applications of Huffman encoding, the goal is to minimize the overall length of the message, and such inefficient scenarios are unlikely. Huffman codes are optimal prefix codes, meaning no code word is a prefix of another, ensuring minimal ambiguity and efficient decoding.n a Huffman encoding, the length of a code word is determined by the frequency of the corresponding symbol. The more frequent a symbol is, the shorter its code word will be. Therefore, the maximal length of a code word in a Huffman encoding is determined by the least frequent symbol in the alphabet.For example, if there are n symbols in the alphabet and the least frequent symbol has a frequency of f, then the maximal length of a code word is f - 1.It is not possible for the codons to be any shorter than three nucleotides because there are 20 different amino acids that need to be encoded. With only two nucleotides per codon, there would only be 16 combinations available, which would not be enough to represent all 20 amino acids.