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Effects like selection in evolution as well as fertility inheritance in thedevelopment of populations can lead to a higher degree of asymmetry inevolutionary trees than expected under a null hypothesis. To identify andquantify such influences, various balance indices were proposed in thephylogenetic literature and have been in use for decades.  However, so far no balance index was based on the number of \emph{symmetrynodes}, even though symmetry nodes play an important role in other areas ofmathematical phylogenetics and despite the fact that symmetry nodes are a quitenatural way to measure balance or symmetry of a given tree.  The aim of this manuscript is thus twofold: First, we will introduce the\emph{symmetry nodes index} as an index for measuring balance of phylogenetictrees and analyze its extremal properties. We also show that this index can becalculated in linear time. This new index turns out to be a generalization of asimple and well-known balance index, namely the \emph{cherry index}, as well asa specialization of another, less established, balance index, namely\emph{Rogers' $J$ index}. Thus, it is the second objective of the presentmanuscript to compare the new symmetry nodes index to these two indices and tounderline its advantages. In order to do so, we will derive some extremalproperties of the cherry index and Rogers' $J$ index along the way and thuscomplement existing studies on these indices. Moreover, we used the programminglanguage \textsf{R} to implement all three indices in the software package\textsf{symmeTree}, which has been made publicly available.

Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties