  Background: The reconstruction of the phylogenetic tree topology of four taxa
is, still nowadays, one of the main challenges in phylogenetics. Its
difficulties lie in considering not too restrictive evolutionary models, and
correctly dealing with the long-branch attraction problem. The correct
reconstruction of 4-taxon trees is crucial for making quartet-based methods
work and being able to recover large phylogenies.
  Results: In this paper we consider an expectation-maximization method for
maximizing the likelihood of (time nonhomogeneous) evolutionary Markov models
on trees. We study its success on reconstructing 4-taxon topologies and its
performance as input method in quartet-based phylogenetic reconstruction
methods such as QFIT and QuartetSuite. Our results show that the method
proposed here outperforms neighbor-joining and the usual (time-homogeneous
continuous-time) maximum likelihood methods on 4-leaved trees with
among-lineage instantaneous rate heterogeneity, and perform similarly to usual
continuous-time maximum-likelihood when data satisfies the assumptions of both
methods.
  Conclusions: The method presented in this paper is well suited for
reconstructing the topology of any number of taxa via quartet-based methods and
is highly accurate, specially regarding largely divergent trees and time
nonhomogeneous data.
