Let $(Zn)_{n\in \mathbb N_0}$ be a Galton-Watson process, i.e.
$$ Z_{n+1} = \sum_{k=1}^{Z_n}\xi_{n,k},\qquad (\xi_{n,k})_{n\in \mathbb N_0,k \in \mathbb N} \quad \text{i.i.d } \mathbb N_0 \text{ valued},$$
where $\xi_{n,k}$ is the offspring distribution of the $k$th individual in the $n$th generation.
Question: What can one say about
$$\mathbb P(Z_n=k), k\in \mathbb N_0 $$
For example, is there an upper bound in terms of $n,k$ and/or $\xi_{n,k}$ on this probability?