Tail asymptotics for the total progeny of the critical killed branching random walk

We consider a branching random walk on $R$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that $P(Z>n)$ is of order $(nln2(n))-1$, which confirms the prediction of Addario-Berry and Broutin [1]