  Quartet trees displayed by larger phylogenetic trees have long been used as
inputs for species tree and supertree reconstruction. Computational constraints
prevent the use of all displayed quartets in many practical problems due to the
number of taxa. We introduce the notion of an Efficient Quartet System (EQS) to
represent a phylogenetic tree with a subset of the quartets displayed by the
tree. We show mathematically that the set of quartets obtained from a tree via
an EQS contains all of the combinatorial information of the tree itself. Using
performance tests on simulated datasets, we also demonstrate that using an EQS
to reduce the number of quartets in pipelines for summary methods of species
tree inference and supertree inference results in only small reductions in
accuracy.
