  Tree balance plays an important role in phylogenetics and other research
areas, which is why several indices to measure tree balance have been
introduced over the years. Nevertheless, a formal definition of what a balance
index actually is and what makes it a useful measure of balance (or, in other
cases, imbalance), has so far not been introduced in the literature. While the
established indices all summarize the (im)balance of a tree in a single number,
they vary in their definitions and underlying principles. It is the aim of the
present manuscript to introduce formal definitions of balance and imbalance
indices that classify desirable properties of such indices and to analyze and
categorize established indices accordingly. In this regard, we review 19
established (im)balance indices from the literature, summarize their general,
statistical and combinatorial properties (where known), prove numerous
additional results and indicate directions for future research by making
explicit open questions and gaps in the literature. We also prove that a few
tree shape statistics that have been used to measure tree balance in the
literature do not fulfill our definition of an (im)balance index, which might
indicate that their properties are not as useful for practical purposes.
Moreover, we show that five additional tree shape statistics from other
contexts actually are tree (im)balance indices according to our definition. The
manuscript is accompanied by the website \url{treebalance.wordpress.com}
containing fact sheets of the discussed indices. Moreover, we introduce the
software package \verb|treebalance| implemented in $\mathsf{R}$ that can be
used to calculate all indices discussed.
