I am told: in a homogeneous branching process , the generating function for the total number of progeny is
$$\phi_0(z) = z\cdot g_0(\phi_0(z))$$
where $g_0(z)$ is the pgf of the first generation.
My problem: I understand the $g_0(\phi_0(z))$ term, since
$$ E(z^{\sum_{i=1}^N Y_{i}}) = g_0(\phi_0(z))$$ where the $Y_i$ is the number of progeny descendant from the $i^{th}$ individual in the first generation (so $N$ must be distributed according to $g_0$).
My question is where does the extra factor of $z$ come from? Is it simply to ensure $\phi_0(0) = 1$? Is there another way to show this?
Thanks!