= Solution
Conditional on $X_n=m$, the next generation is the sum of $m$ independent offspring counts, each with <probability generating function> $F_1(s)$. Hence
$$
E[s^{X_{n+1}}\mid X_n=m]=F_1(s)^m.
$$
This also holds for $m=0$, when the empty sum is zero and the empty product is one. The <law of total expectation> gives
$$
\boxed{F_{n+1}(s)=E[F_1(s)^{X_n}]=F_n(F_1(s)).}
$$
For complex $|s|\leq1$, the expectations are absolutely bounded because $|F_1(s)|\leq1$. This proves the <branching-process generating-function iteration>, with $F_0(s)=s$.
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