An estimation of probabilites of large deviations for a critical Galton–Watson process

Let $Z_n$, $n=0,1,\dots,$ be a critical Galton–Watson process with $Z_0=1$. An estimation of $\mathbf P(Z_n>k)$ is obtained for every $k>0$ under the assumption that $\mathbf P(Z_1>k)<e^{-\alpha k}$, $\alpha>0$.