Simple models for scaling in phylogenetic trees

Many processes and models produce trees with depth scaling 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 depth scaling. For the first one, Ford\\\'s alpha model, power-law scaling in the depth was established analytically. Our numerical results illustrate that the asymptotic regime is approached only at very large tree sizes. A second model, the activity model, is introduced here. We show analytically and numerically that its depth also displays power-law scaling at a critical parameter value.

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