Abstract
We introduce a one-parametric family of tree growth models, in which branching probabilities decrease with branch age $\ensuremath{\tau}$ as ${\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}}$. Depending on the exponent $\ensuremath{\alpha}$, the scaling of tree depth with tree size $n$ displays a transition between the logarithmic scaling of random trees and an algebraic growth. At the transition $(\ensuremath{\alpha}=1)$ tree depth grows as ${(logn)}^{2}$. This anomalous scaling is in good agreement with the trend observed in evolution of biological species, thus providing a theoretical support for age-dependent speciation and associating it to the occurrence of a critical point.