I have a time-inhomogeneous Galton-Watson branching process over a finite number of generations $n$. I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous, but I can compute the expected number of survivors $\mu_i$ at each level. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!