  We introduce and study the class of branching-stable point measures, which
can be seen as an analog of stable random variables when the branching
mechanism for point measures replaces the usual addition. In contrast with the
classical theory of stable (L\'evy) processes, there exists a rich family of
branching-stable point measures with \emph{negative} scaling exponent, which
can be described as certain Crump-Mode-Jagers branching processes. We
investigate the asymptotic behavior of their cumulative distribution functions,
that is, the number of atoms in $(-\infty, x]$ as $x\to \infty$, and further
depict the genealogical lineage of typical atoms. For both results, we rely
crucially on the work of Biggins.
