Simple models for scaling in phylogenetic trees

Many processes and models --in biological, physical, social, and other contexts-- produce trees whose depth scales logarithmically
with the number of leaves. Phylogenetic trees, describing the
evolutionary relationships between biological species, are
examples of trees for which such scaling is not observed. With
this motivation, we analyze numerically two branching models
leading to non-logarithmic scaling of the depth with the number of
leaves. For Ford\'s alpha model, although a power-law scaling of
the depth with tree size was established analytically, our
numerical results illustrate that the asymptotic regime is
approached only at very large tree sizes. We introduce here a new
model, the activity model, showing analytically and numerically that it also displays a power-law scaling of the depth
with tree size at a critical parameter value.