Age-dependent and lineage-dependent speciation and extinction in the imbalance of phylogenetic trees

It is known that phylogenetic trees are more imbalanced than expected from a birth–death model with constant rates of speciation and extinction, and also that imbalance can be better fit by allowing the rate of speciation to decrease as the age of the parent species increases. If imbalance is measured in more detail, at nodes within trees as a function of the number of species descended from the nodes, age-dependent models predict levels of imbalance comparable to real trees for small numbers of descendent species, but predicted imbalance approaches an asymptote not found in real trees as the number of descendent species becomes large. Age-dependence must therefore be complemented by another process such as inheritance of different rates along different lineages, which is known to predict insufficient imbalance at nodes with few descendent species, but can predict increasing imbalance with increasing numbers of descendent species.