Central Moments and Probability Distributions of Three Measures of Phylogenetic Tree Imbalance

Several recent studies have included attempts to use the balance of phylogenetic trees, i.e., the extent to which sister groups within a tree tend to be the same size, to test hypotheses about the macroevolutionary processes that produced them. Such tests require measures of balance or imbalance and the moments or probability distributions of these measures under some null models. In earlier work, I developed recursion equations for the mean, variance, skewness, and complete probability distribution of Colless's coefficient of imbalance (I) (Rogers, 1994, Evolution 48:2026–2036). In this paper, I report the extension of these techniques to two additional imbalance measures, the number of unbalanced nodes on a tree (J) and Sackin's index (K), under both the equal-rates Markov (ERM) model and the equal probability (EP) model. I also show how to find the correlations and joint probability distributions of all pairs of these three coefficients. I and K are so highly correlated for trees of all sizes that K contains little additional information about tree balance that is not conveyed by I. The correlation of I and J, however, decreases rapidly with increasing tree size, indicating that the testing of macroevolutionary hypotheses may be refined by employing the joint distribution of these two coefficients. The results of two simulation studies of non-ERM speciation processes are used to illustrate how the joint distribution of I and J may be used