  We consider population-size-dependent branching processes (PSDBPs) which
eventually become extinct with probability one. For these processes, we derive
maximum likelihood estimators for the mean number of offspring born to
individuals when the current population size is $z\geq 1$. As is standard in
branching process theory, an asymptotic analysis of the estimators requires us
to condition on non-extinction up to a finite generation $n$ and let
$n\to\infty$; however, because the processes become extinct with probability
one, we are able to demonstrate that our estimators do not satisfy the
classical consistency property ($C$-consistency). This leads us to define the
concept of $Q$-consistency, and we prove that our estimators are $Q$-consistent
and asymptotically normal. To investigate the circumstances in which a
$C$-consistent estimator is preferable to a $Q$-consistent estimator, we then
provide two $C$-consistent estimators for subcritical Galton-Watson branching
processes. Our results rely on a combination of linear operator theory,
coupling arguments, and martingale methods.
