  Nonhierarchical clustering depending on unsupervised algorithms may not
retrieve the optimal partition of datasets. Determining if clusters fit
``natural partitions`` can be achieved using cluster validity indices (CVIs).
Most existing CVIs consider criteria such as cohesion, separation, and their
equivalents. However, these binary relations may provide neither the optimal
measure of partition suitability nor reference values corresponding to the
worst partition. Moreover, previous CVI studies have been mostly focused on
fitting correct partitions according to researchers' a priori assumptions. In
contrast, we investigated desirable properties of CVIs, namely, scale shift
transform invariance, optimal clustering, and unbiased clustering with
representing the worst partition. Then, we conducted experiments to evaluate
whether existing CVIs fulfill these properties. As none of these CVIs fulfilled
the desired properties, we propose the simplicity index, which measures the
simplicity of tree structures in clusters. The simplicity index is the unique
index invariant to the ``correct rate`` and provides both a reference
indicating the most complex partition and the best value indicating the
simplest one.
