Tree shape statistics, particularly measures of tree (im)balance, play an important role in the analysis of the shape of phylogenetic trees. With applications ranging from testing evolutionary models to studying the impact of fertility inheritance and selection, or tumour development and language evolution, the assessment of measures of tree balance is important. Currently, a multitude of at least 30 (im)balance indices can be found in the literature, alongside numerous other tree shape statistics. This diversity prompts essential questions: how can we assist researchers in choosing only a small number of indices to mitigate the challenges of multiple testing? Is there a preeminent balance index tailored to specific tasks? This research expands previous studies on the examination of index power, encompassing almost all established indices and a broader array of alternative models, such as a variety of trait-based models. Our investigation reveals distinct groups of balance indices better suited for different tree models, suggesting that decisions on balance index selection can be enhanced with prior knowledge. Furthermore, we present the R software package poweRbal which allows the inclusion of new indices and models, thus facilitating future research on the power of tree shape statistics.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.