Branching-process generating-function iteration ($G_{n+1}=G\circ G_n,G_0(s)=s$)

Conditioning on the initial individual's children gives independent copies of the population $n$ generations later. Their PGFs multiply, and averaging the product over the offspring number yields $G(G_n(s))$. Conditioning on the last generation gives $G_n(G(s))$ as well; both agree because $G_n$ is the $n$fold iterate of $G$.