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.