= Finite-horizon total progeny generating function
{title2=$H_{n+1}(s)=sG(H_n(s)),\quad H_0(s)=1$}
Count the ancestor and the next $n-1$ generations of a <Galton-Watson process>. Conditioning on the ancestor's offspring, their independent descendant trees each have the preceding finite-horizon <probability generating function>, proving the displayed recursion. If the count instead excludes the ancestor, the recursion is $\widehat H_{n+1}(s)=G(s\widehat H_n(s))$ with $\widehat H_0=1$. The indexing convention must be specified.
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