  Let $(Z_n)$ be a supercritical branching process in a random environment $\xi
= (\xi_n)$. We establish a Berry-Esseen bound and a Cram\'er's type large
deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also
improve some earlier results about the harmonic moments of the limit variable
$W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_{\xi} Z_n$ is the
normalized population size.
