Phylogenetic trees serve to represent phylogenetic relationships arising from evolutionary events such as speciation and extinction. In these binary trees, leave nodes represent observed (extant) species while inner nodes stand for events of speciation. The shape of a tree is its pure graph structure neglecting the annotation of evolutionary time and species. Tree shapes based on empirical deviate significantly from those predicted by completely uncorrelated speciation processes such as the Equal Rates Markov (ERM) model. In this model, the depth (average distance of leaves from root) scales logarithmically with the number of leaves. A faster depth scaling with system size is observed in the real trees as a sign of systematic imbalance. Here we introduce a potential explanation of tree imbalance in terms of a speciation rate that decreases with the age of a species. Trees are grown by iterative branching. The branching leaf is chosen with probability inversely proportional to the time that passed since the leaf's creation. This model produces trees with depth scaling as log2 of tree size, in good agreement with trees in the databases TREEBASE and PANDIT. Furthermore, the likelihood of this model with respect to real trees is typically larger or equal to that of the AB model [Aldous, 1996]. The latter model is known to account for the observed imbalance but is not based on any evolutionary mechanism [Blum & Francois, Syst. Biol., 2006]. Along with the modeling results, we introduce a general efficient Monte-Carlo method for likelihood estimation of growing tree models that do not factorize over branches. The method will facilitate verification and comparison of many complex growth models using large real trees.