= Solution
For a <Galton-Watson process>, condition on the number $K$ of children of the initial individual. The descendants of these children over the next $n$ generations are independent copies of $X_n$. The <probability generating function> of their total, conditional on $K=k$, is $G_n(s)^k$. Averaging over the family-size law gives
$$
G_{n+1}(s)=\mathbb E[G_n(s)^K]=\boxed{G(G_n(s))},\qquad G_0(s)=s.
$$
This is the <branching-process generating-function iteration>. Conditioning instead on the last generation gives $G_n(G(s))$; both orders agree because $G_n$ is the $n$th iterate of the same function $G$. The argument holds on the usual <PGF> domain $|s|\leq1$, with larger domains available when the corresponding expectations converge.
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