Semi‐strict supertrees

Abstract

A method to calculate semi-strict supertrees is proposed. The semi-strict supertrees are calculated by creating the matrix that represents all the groups in the source trees (as done in already existing techniques), and then finding the trees determined by the ultra-clique. The ultra-clique is defined as the set of characters where each possible subset is compatible with each possible subset from the entire matrix. Finding the ultra-clique is computationally complex (since in most cases many of the characters have missing entries), but a heuristic method yields reliable results. When the trees have no conflict, or when there are only two trees, the method produces the exact result for any ordering of the input trees and any ordering of the groups within them; when there are more than two trees and they have conflict, a single ordering or sequence can create some spurious groups, but doing multiple sequences eliminates the spurious groups. The method uses only state set operations, and is thus easily implemented in computer programs. Unlike any existing type of supertree, semi-strict supertrees display all the groups, and only those groups, that are implied by at least some combination of the input trees and contradicted by none. The idea that supertrees should take into account the number of occurences of a given group, so as to retain some groups even in the case of conflict, is discussed; it is argued that a conceptual equivalent of the majority rule consensus is not possible when the sets of taxa differ among trees. Also, when pruning taxa from a set of trees, the supertree can display groups that contradict the consensus for the entire trees, suggesting that supertrees for matrices with very dissimilar sets of taxa should be interpreted with caution. If (for any valid reason) the data cannot be combined in a single matrix, it is advisable that the taxon sets in the matrices be as similar as possible.