Further convergence results for entropies of random branching processes

Suppose a supercritical Galton-Watson random branching process with a finite number of types is given. Then, with probability one in the set of infinitely extended derivation trees, functions of derivations that are additive on subtrees normalized by the number of subtrees converge to their expected values. As corollaries of this general result, convergence is shown for sample entropy, codeword length for codes assigned to subtrees, and number of terminals. An extension of this result shows convergence of ratios of such functions yielding convergence of entropy per terminal as a special case