Consider a Galton-Watson tree with offspring distribution $\text{B}(n,p)$, for some constant $n$ and $p$. Let $a_i$ denote the number of vertices on the $i$'th level of the tree. It then holds that:

$a_0=1$

$a_{i+1}\sim \text{B}(n\cdot a_i,p)$

I am looking for a probabilistic upper bound on the tree's growth. For example, a Theorem that gives an upper bound on $\Pr(a_i > c\cdot (np)^i)$, for $c>1$, would be nice.

My guess is that something like this should be known, but I haven't managed to locate it so far. I would be thankful if anyone can point me where to look.