Expected and minimal values of a universal tree balance index

Although the analysis of rooted tree shape has wide-ranging applications, notions of tree balance have developed independently in different domains. In computer science, a balanced tree is one that enables efficient updating and retrieval of data, whereas in biology tree balance quantifies bias in evolutionary processes. The lack of a precise connection between these concepts has stymied the development of universal indices and general results. We recently introduced a new tree balance index, $J^1$, that, unlike prior indices popular among biologists, permits meaningful comparison of trees with arbitrary degree distributions and node sizes. Here we explain how our new index generalizes a concept that underlies the definition of the weight-balanced tree, an important type of self-balancing binary search tree. Our index thus unifies the tree balance concepts of biology and computer science. We provide new analytical results to support applications of this universal index. First, we quantify the accuracy of approximations to the expected values of $J^1$ under two important null models: the Yule process and the uniform model. Second, we investigate minimal values of our index. These results help establish $J^1$ as a universal, cross-disciplinary index of tree balance that generalizes and supersedes prior approaches.