  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.

Quantitative Methods;Statistical Mechanics;Populations and Evolution