The information content of trees and their matrix representations.

Any tree can be represented by a variety of matricesfrom which the tree can be reconstructed. Such matrixrepresentations (encodings) of trees have been used in anumber of contexts, including measuring tree similarity(Farris, 1973), cophylogeny (Brooks, 1981), and consen-sus (Nelson and Ladiges, 1994b; Phillips and Warnow,1996),buttheyhavebecomemoreprominentthroughthematrix representation with parsimony (MRP) approachto supertree construction (Baum, 1992; Ragan, 1992;Sanderson et al., 1998). Most published supertrees haveutilized MRP (e.g., Jones et al., 2002; Kennedy and Page,2002; Pisani et al., 2002; Purvis, 1995a), in which matrixrepresentations of trees are combined into a compositematrix and analyzed with parsimony. In standard MRP,the elements of matrix representations are the relation-ships given by the full splits or bipartitions of leaves (ter-minal taxa) present in the trees. In the context of rootedtrees, standard matrix representations have one binary(pseudo)characterforeachcladeinwhichclademembersarescoredconventionally(andarbitrarily)as‘1’andnon-members and the root (MRP outgroup) as ‘0’ (Table 1). Purvis (1995b) developed an alternative matrix rep-resentation (Purvis coding) intended to compensate forapparent overweighting of larger trees in standard MRP.His method, which is applicable only to rooted trees, at-tempted to remove redundant information in standardmatrix representations, but was criticized by Ronquist(1996)whoarguedthattheinformationremovedwasnotredundant. Williams and Humphries (2003) advocatedthe use of matrix representations based on three-itemstatements or triplets (Nelson and Ladiges, 1992, 1994b;Nelson and Platnick, 1991; Wilkinson et al., 2001), whichare also applicable only to rooted trees, and they usedsuch matrices, and an associated differential weighting,as a yardstick to compare and contrast standard andPurvis matrix representations. Matrix representationscan also be based on pairwise pathlength distances be-tween leaves (Lapointe and Cucumel, 1997; Lapointe etal., 2003) and nonbinary discrete characters (Semple andSteel, 2002). We believe the literature on the relative merits of al-ternative binary matrix representations of trees to beconfused and confusing. Here we use a quantitativemeasure, cladistic information content (CIC; Thorleyet al., 1998), to compare the information content of treesand their matrix representations. Speciﬁcally, we revisitRonquist’s (1996) critique of Purvis coding. We demon-strate that the information removed by this method isredundant in the sense that Purvis (1995b) intended, butthat it does not remove all such redundant informationasPurvissupposed.BuildingonRonquist(1996),wedis-cuss the distinction between the representation of a treeand the representation of the data from which a tree hasbeen inferred and its importance in choosing a matrixrepresentation. This leads us to consider logical relationsand dependencies among pseudocharacters from thesametree,andhighlightfundamentalinferencerulesthatform the basis of the logical calculus of cladistic relation-ships. We show that fractional weighting as proposedby Nelson and Ladiges (1992), and the form of this usedby Williams and Humphries (2003), does not account forall logical dependencies between triplets. This leads usto propose an alternative weighting scheme. Finally, wereview the results and present a critical discussion ofWilliams and Humphries’ (2003) comparison of alterna-tive methods. We use the terms triplet and three-item