Novel Information Theory-Based Measures for Quantifying Incongruence among Phylogenetic Trees

Phylogenies inferred from different data matrices often conflict with each other necessitating the development of measures that quantify this incongruence. Here, we introduce novel measures that use information theory to quantify the degree of conflict or incongruence among all nontrivial bipartitions present in a set of trees. The first measure, internode certainty (IC), calculates the degree of certainty for a given internode by considering the frequency of the bipartition defined by the internode (internal branch) in a given set of trees jointly with that of the most prevalent conflicting bipartition in the same tree set. The second measure, IC All (ICA), calculates the degree of certainty for a given internode by considering the frequency of the bipartition defined by the internode in a given set of trees in conjunction with that of all conflicting bipartitions in the same underlying tree set. Finally, the tree certainty (TC) and TC All (TCA) measures are the sum of IC and ICA values across all internodes of a phylogeny, respectively. IC, ICA, TC, and TCA can be calculated from different types of data that contain nontrivial bipartitions, including from bootstrap replicate trees to gene trees or individual characters. Given a set of phylogenetic trees, the IC and ICA values of a given internode reflect its specific degree of incongruence, and the TC and TCA values describe the global degree of incongruence between trees in the set. All four measures are implemented and freely available in version 8.0.0 and subsequent versions of the widely used program RAxML.