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Adapting a definition given by Bjerkevik and Lesnick for multiparameterpersistence modules, we introduce an $\ell^p$-type extension of theinterleaving distance on merge trees. We show that our distance is a metric,and that it upper-bounds the $p$-Wasserstein distance between the associatedbarcodes. For each $p\in[1,\infty]$, we prove that this distance is stable withrespect 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 mergetrees.

The Universal $\ell^p$-Metric on Merge Trees