Branching Process
if initial population $0$ is equal to $1$, i.e. $X_0 \equiv 1$
and $S_k$ is the total population size up to generation $k$ meaning $S_k = 1 +X_1 +...+X_k$.
If $G(z)$ is the Probability Generating Function of the number of offspring produced by a single member.
and $H_k(z) $ is the The Probability Generating Function of $S_k$.
what is the relationship between $H_k(z)$ and $H_{k-1}(z)$
My calculation gave me the following relationship, but I am not sure:
$$H_k(z) = G(H_{k-1}(z))$$