  Take a continuous-time Galton-Watson tree. If the system survives until a
large time $T$, then choose $k$ particles uniformly from those alive. What does
the ancestral tree drawn out by these $k$ particles look like? Some special
cases are known but we give a more complete answer. We concentrate on
near-critical cases where the mean number of offspring is $1+\mu/T$ for some
$\mu\in\mathbb{R}$, and show that a scaling limit exists as $T\to\infty$.
Viewed backwards in time, the resulting coalescent process is topologically
equivalent to Kingman's coalescent, but the times of coalescence have an
interesting and highly non-trivial structure. The randomly fluctuating
population size, as opposed to constant size populations where the Kingman
coalescent more usually arises, have a pronounced effect on both the results
and the method of proof required. We give explicit formulas for the
distribution of the coalescent times, as well as a construction of the
genealogical tree involving a mixture of independent and identically
distributed random variables. In general subcritical and supercritical cases it
is not possible to give such explicit formulas, but we highlight the special
case of birth-death processes.
