Performance of a new invariants method on homogeneous and   non-homogeneous quartet trees
  An attempt to use phylogenetic invariants for tree reconstruction was made at the end of the 80s and the beginning of the 90s by several authors (the initial idea due to Lake and Cavender and Felsenstein in 1987. However, the efficiency of methods based on invariants is still in doubt, probably because these methods only used few generators of the set of phylogenetic invariants. The method studied in this paper was first introduced by Casanellas, Garcia and Sullivant and it is the first method based on invariants that uses the whole set of generators for DNA data. The simulation studies performed in this paper prove that it is a very competitive and highly efficient phylogenetic reconstruction method, especially for non-homogeneous models on phylogenetic trees. 
Marta Casanellas, Jesus Fernandez-Sanchez