  We introduce a one-parametric family of tree growth models, in which
branching probabilities decrease with branch age $\tau$ as $\tau^{-\alpha}$.
Depending on the exponent $\alpha$, 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 ($\alpha=1$) tree depth grows as $(\log
n)^2$. 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.
