A finitely stable edit distance for merge trees

In this paper, we defined a novel edit distance for merge trees, which we argued to be suitable for a broad range of applications. Relying also on some technical results contained in other works, we investigated its stability properties, which ended up being analogous to the ones of the 1-Wasserstein distance between persistence diagrams. We tested and compared our metric against the interleaving distance in several simulations and case studies, highlighting the trade-off between stability and sensitivity when choosing the appropriate metric for a given data analysis problem, much alike the bias-variance trade-off in statistical modeling. In the appendix, we also compared our metric with other edit distances appearing in the literature, with both theoretic and practical considerations.