  Kingman's coalescent is a random tree that arises from classical population
genetic models such as the Moran model. The individuals alive in these models
correspond to the leaves in the tree and the following two laws of large
numbers concerning the structure of the tree-top are well-known: (i) The
(shortest) distance, denoted by $T_n$, from the tree-top to the level when
there are $n$ lines in the tree satisfies $nT_n \xrightarrow{n\to\infty} 2$
almost surely; (ii) At time $T_n$, the population is naturally partitioned in
exactly $n$ families where individuals belong to the same family if they have a
common ancestor at time $T_n$ in the past. If $F_{i,n}$ denotes the size of the
$i$th family, then $n(F_{1,n}^2 + \cdots + F_{n,n}^2) \xrightarrow{n\to
\infty}2$ almost surely. For both laws of large numbers we prove corresponding
large deviations results. For (i), the rate of the large deviations is $n$ and
we can give the rate function explicitly. For (ii), the rate is $n$ for
downwards deviations and $\sqrt n$ for upwards deviations. For both cases we
give the exact rate function.
