Consider the following critical Galton-Watson process: initially there is a population of $Z(0) = z_0$. The distribution of children for each node follows a binomial law, with maximum value $d$; i.e. the probability that a node has $k$ children is $$ p_k = \binom{d}{k} q^k (1-q)^k $$ where $q = 1/d$.
In this case, the expected number of children is $1$, so this is a critical process.
Now, my question is the following: define the random variable $S = \sum_{i=0}^{\infty} Z(i)$, the total number of nodes in the full process. With probability one we have $S < \infty$. What can we say about the distribution of $S$? That is, can one estimate $P(S = i)$?
I have some preliminary calculations that suggest that for $z_0 = 1$ we have $P(S = i) \approx i^{-3/2}$. I don't have a proof of this and I am not sure what happens when $z_0$ becomes large.
Thanks for help or pointers