  In this article, we define a new non-archimedean metric structure, called
cophenetic metric, on persistent homology classes of all degrees. We then show
that zeroth persistent homology together with the cophenetic metric and
hierarchical clustering algorithms with a number of different metrics do
deliver statistically verifiable commensurate topological information based on
experimental results we obtained on different datasets. We also observe that
the resulting clusters coming from cophenetic distance do shine in terms of
different evaluation measures such as silhouette score and the Rand index.
Moreover, since the cophenetic metric is defined for all homology degrees, one
can now display the inter-relations of persistent homology classes in all
degrees via rooted trees.
