  We consider a supercritical branching process $(Z_n)$ in a random environment
$\xi$. Let $W$ be the limit of the normalized population size
$W_n=Z_n/E[Z_n|\xi]$. We first show a necessary and sufficient condition for
the quenched $L^p$ ($p>1$) convergence of $(W_n)$, which completes the known
result for the annealed $L^p$ convergence. We then show that the convergence
rate is exponential, and we find the maximal value of $\rho>1$ such that
$\rho^n(W-W_n)\rightarrow 0$ in $L^p$, in both quenched and annealed sense.
Similar results are also shown for a branching process in a varying
environment.
