  Effects like selection in evolution as well as fertility inheritance in the
development of populations can lead to a higher degree of asymmetry in
evolutionary trees than expected under a null hypothesis. To identify and
quantify such influences, various balance indices were proposed in the
phylogenetic literature and have been in use for decades.
  However, so far no balance index was based on the number of \emph{symmetry
nodes}, even though symmetry nodes play an important role in other areas of
mathematical phylogenetics and despite the fact that symmetry nodes are a quite
natural 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 phylogenetic
trees and analyze its extremal properties. We also show that this index can be
calculated in linear time. This new index turns out to be a generalization of a
simple and well-known balance index, namely the \emph{cherry index}, as well as
a specialization of another, less established, balance index, namely
\emph{Rogers' $J$ index}. Thus, it is the second objective of the present
manuscript to compare the new symmetry nodes index to these two indices and to
underline its advantages. In order to do so, we will derive some extremal
properties of the cherry index and Rogers' $J$ index along the way and thus
complement existing studies on these indices. Moreover, we used the programming
language \textsf{R} to implement all three indices in the software package
\textsf{symmeTree}, which has been made publicly available.
