Less established tree shape statistics that are (im)balance indices

Abstract

In this chapter, we consider nine tree shape statistics, namely the total internal path length, the total path length and its normalized version the average vertex depth, the maximum depth, the maximum width, the modified maximum difference in widths, and the maximum width over maximum depth, as well as the tree shape statistics known as stairs1 and stairs2. These have occurred in different contexts, e.g., in the identification of disease transmission patterns [25] and the study of the evolutionary history of viral populations [109], but so far have not been considered as (im)balance indices. Since they have not been thoroughly studied yet, we now briefly explore some of their properties. Most importantly, we show that all nine of them fulfill our definition of a balance or imbalance index, results that – to the best of our knowledge – are new to the literature. An in-depth analysis of these tree shape statistics is beyond the scope of this book, but we anticipate and hope that the short fact sheets we present on these indices will inspire new research.