 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 givesThis 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 $|s|\le 1$, with larger domains available when the corresponding expectations converge.