Phylogenetic trees describe the relationships between species in the evolutionary process, and provide information about the rates of diversification. To understand the mechanisms behind macroevolution, we consider a class of multitype branching processes called Markovian binary trees (MBTs). MBTs allow for trait-based variation in diversification rates, and provide a flexible and realistic probabilistic model for phylogenetic trees. We develop an approximate Bayesian computation (ABC) scheme to infer the rates of MBT parameters by exploiting the information in the shapes of phylogenetic trees. We evaluate the accuracy of this inference method using simulation studies, and find that our method is able to detect variation in the diversification rates, with accuracy comparable to, and generally better than, likelihood-based methods. In an application to a real-life phylogeny of squamata, we reinforce conclusions drawn from earlier studies, in particular supporting the existence of ovi-/viviparity transitions in both directions. Our method demonstrates the potential for more complex models of evolution to be employed in phylogenetic inference, in conjunction with likelihood-free schemes.