A height-based metaconcept for rooted tree balance and its implications for the $B_1$ index

Tree balance has received considerable attention in recent years, both in phylogenetics and in other areas. Numerous (im)balance indices have been proposed to quantify the (im)balance of rooted trees. A recent comprehensive survey summarized this literature and showed that many existing indices are based on similar underlying principles. To unify these approaches, three general metaconcepts were introduced, providing a framework to classify, analyze, and extend imbalance indices. In this context, a metaconcept is a function $\Phi_f$ that depends on another function $f$ capturing some aspect of tree shape. In this manuscript, we extend this line of research by introducing a new metaconcept based on the heights of the pending subtrees of all inner vertices. We provide a thorough analysis of this metaconcept and use it to answer open questions concerning the well-known $B_1$ balance index. In particular, we characterize the tree shapes that maximize the $B_1$ index in two cases: (i) arbitrary rooted trees and (ii) binary rooted trees. For both cases, we also determine the corresponding maximum values of the index. Finally, while the $B_1$ index is induced by a so-called third-order metaconcept, we explicitly introduce three new (im)balance indices derived from the first- and second-order height metaconcepts, respectively, thereby demonstrating that pending subtree heights give rise to a variety of novel (im)balance indices.