I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which may or may not survive. This branching process goes through $n$ levels, so there are $2^n$ leaf nodes. I want to estimate the probability that there is at least one surviving leaf node.

For each level $i = 0, \dots, n$ I have a lower bound $P_i$ on the probability that any given level-$i$ node survives. However, all of these events may be very dependent on each other.

I also have a lower bound $Q$ on the probability that any given level-$i$ node survives to level $i+1$ (independent of any other events that may have occured).

I have $2^n P_n \ll 1$, and I want to show that the probability that some leaf node survives is about $2^n P_n$. At the very least, I would like to show that the probability of survival is much more than $2^n Q^n$. This is what would occur if I had a uniform Galton-Watson process with probability $Q$ exactly.

Intuitively, the worst possible distribution of probabilities would be if all the nodes at level $n$ had completely the same support. In other words, there is an initial probability of $P_n$ that all the leaves in the tree survive. But we can rule this out, because in this case the probability that a level-1 node survives is also $P_n$, and $P_1 \gg P_n$.

Thanks for any help