*Author to whom correspondence should be addressed. E-mail: a.purvis@ic.ac.uk J. theor. Biol. (2002) 214, 99}103 doi:10.1006/jtbi.2001.2443, available online at http://www.idealibrary.com on Evaluating Phylogenetic Tree Shape: Two Modi5cations to Fusco & Cronk:s Method ANDY PURVIS*,ARIS KATZOURAKIS AND PAUL-MICHAEL AGAPOW Department of Biological Sciences, Imperial College, Silwood Park, Ascot, Berkshire S¸57P>, ;.K. (Received on 14 June 2001, Accepted in revised form on 28 August 2001) Measurement of the degree of asymmetry in phylogenetic trees is important because a tree's shape re#ects the process by which it has grown. For example, highly asymmetric trees are evidence that species have had di!erent potential for diversi"cation. Of the tree shape measures in the literature, that proposed by Fusco & Cronk (J. theor. Biol. 175, 235}243) appears to be particularly useful, because it does not require fully-resolved trees whose terminals are of equal taxonomic rank. The value of the asymmetry or imbalance at a node is intended to be independent of the number of species ultimately descended from the node. In this paper, however, we point out that the value does depend upon species number. We propose two modi"cations that remove the dependency and so increase the measure's useful- ness. We illustrate the use of the modi"ed measures, which are implemented in a freely- available program, MESA.  2002 Academic Press The shape of a phylogenetic tree contains valu- able information about the processes under which the tree has grown (Harvey et al., 1996). Many statistics are available that capture various aspects of tree topology (see Mooers & Heard, 1997, for a review). However, most can be used only with completely resolved estimated trees, so cannot be used with the great majority of pub- lished phylogenies that have polytomies (nodes with more than two daughter branches) repre- senting uncertainty about the underlying struc- ture. Fusco & Cronk (1995) proposed a measure of tree asymmetry or imbalance with two unique advantages. First, it can be used with incomplete- ly resolved trees. Second, it can be used with trees whose terminal taxa are not of equal taxonomic rank, so long as the species-richnesses are known. Furthermore, Fusco & Cronk (1995) claimed that the imbalance, I, of a node is independent of the number of species, S, ultimately descending from it under a null model of clade growth in which contemporaneous lineages have the same e!ec- tive speciation rate (the Markovian null model: Raup et al., 1973). Such constancy of expectation would faciliate the use of I in assessing a tree's compatibility with the null model, and in com- paring the shape of multiple trees. In this note, we show that the expected value of I is not in fact independent of S. We brie#y discuss the e!ect of this dependency on tree shape measurements and comparisons, propose two simple modi"cations that remove the dependency, and illustrate their use. The modi"ed statistics are implemented in a freely available program, MESA. I is calculated as follows (Fusco & Cronk, 1995). Consider a bifurcation in the phylogeny for which S'3 (nodes with S"3 lack alterna- tive topologies). Then, for that bifurcation: I"(B!m)/(M!m), 0022}5193/01/010099#05 $35.00/0  2002 Academic Press