Kraft-McMillan inequality

By induction on construction, prove that for a binary Huffman code the sum in the Kraft-McMillan inequality is exactly equal to one.

We need to construct a uniquely decodable binary homomorphism from the alphabet $\left\{ a, b, c, d, e, f, g, h\right\}$. For certain reasons, it has been decided to use code words of lengths $5,1,3,3,4,8$ for the symbols $a, b, c, d, e, f$ respectively. It is known that the average frequencies of the symbols $g$ and $h$ are $\frac{1}{8}$ and $\frac{1}{5}$ respectively. Find the optimal lengths of the code words for $g$ and $h$.

By induction on the construction, prove that for a binary Huffman code, the sum in the Kraft-McMillan inequality is exactly equal to one.

Prove that for any optimal binary homomorphism, the sum in the Kraft-McMillan inequality is exactly equal to one. Demonstrate that for non-binary homomorphisms this may fail.

We need to construct an unambiguous binary homomorphism from the alphabet $\left\{ a, b, c, d, e, f, g, h\right\}$. For certain reasons, for the symbols $a, b, c, d, e, f$ it has been decided to use code words of lengths $5,1,3,3,4,8$ respectively. It is known that the average frequencies of the symbols $g$ and $h$ are $\frac{1}{8}$ and $\frac{1}{5}$ respectively. Find the optimal lengths of the code words for $g$ and $h$.