  Earthquakes in seismological catalogs and acoustic emission events in lab
experiments can be statistically described as a linear Hawkes point process,
where the spatio-temporal rate of events is a linear superposition of
background intensity and the aftershock clusters triggered by preceding
activity. Traditionally, statistical seismology has interpreted this model as
the outcome of an epidemic branching process, where one-to-one causal links can
be established between mainshocks and aftershocks. Declustering techniques have
been used to infer the underlying triggering trees and relate their topological
properties with epidemic branching models. Here, we review how the standard
Epidemic Type Aftershock Sequence (ETAS) model extends from the Galton-Watson
(GW) branching processes and bridges two extreme cases: Poisson sampling and
scale-free power-law trees. We report the most essential topological properties
expected in GW epidemic trees: the branching probability, the distribution of
tree size, the expected family size, and the relation between average
leaf-depth and tree size. We find that such topological properties depend
exclusively on two sampling parameters of the standard ETAS model: the average
branching ratio $N_b$ and the exponent ratio $\alpha/b$ determining the
branching probability distribution. From these results, one can use the
memory-less GW as a null-model for empirical triggering processes and assess
the validity of the ETAS model to reproduce the statistics of natural and
artificial catalogs.
