  Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. Consider the
Lotka-Nagaev estimator for the offspring mean. In this paper, we establish
self-normalized Cram\'{e}r type moderate deviations and Berry-Esseen's bounds
for the Lotka-Nagaev estimator. The results are believed to be optimal or near
optimal.
