In a standard Galton-Watson process, you have the population in each generation given by $Z_n=\sum_{j=1}^{Z_{n-1}} X_j$. And the $X_j$s are all i.i.d. random variables. As far as I can tell, their i.i.d.-ness holds not just in a single generation but across all generations, so the $X_j$s in the 1000th generation are independent of those in the 10th generation, and distributed the same way. (Is that right?)
For that standard Galton-Watson process, the total progeny $T$ (total number of people who ever live from $n=0$ onwards) satisfies: $P(T=a | Z_0=b)=\frac{b}{a} P(Z_1 = a-b | Z_0=a)$.
Or, if we let $\varphi_{T}(s)$ be the probability generating function for $T$, and $\varphi$ be the p.g.f. for the distribution for each $X_j$, then $\varphi_{T}(s)=s \cdot \varphi (\varphi_{T}(s))$.
But do both of those facts still hold if we let the $X_j$s have different distributions in each generation? Specifically, if their distributions also depend on the size of the previous generation--if it's actually a population-size-dependent Galton-Watson process. For instance, the $X_j$s may have higher probability of returning 0 or 1 if the population is already very big. (This sort of process is discussed in these papers, but they don't look at total progeny.)
I've gone through the proofs of both (the one from the original Dwass paper for the first, and the basic proof by induction for the second) and I can't tell for sure whether they'd also work with the $X_j$ distributions being dependent on population size. (I realise that, for the second, $\varphi$ would have to be replaced with something like the limit of the offspring distribution as population size approaches 0 or $\infty$.) Do the results still hold?
Bonus question: are there any papers out there that talk about total progeny for population-size-dependent branching processes? I haven't had much luck finding any.