# The Universal $ell^p$-Metric on Merge Trees. (arXiv:2112.12165v1 [cs.CG])

Adapting a definition given by Bjerkevik and Lesnick for multiparameter
persistence modules, we introduce an $ell^p$-type extension of the
interleaving distance on merge trees. We show that our distance is a metric,
and that it upper-bounds the $p$-Wasserstein distance between the associated
barcodes. For each $pin[1,infty]$, we prove that this distance is stable with
respect to cellular sublevel filtrations and that it is the universal (i.e.,
largest) distance satisfying this stability property. In the $p=infty$ case,
this gives a novel proof of universality for the interleaving distance on merge
trees.