A Persistence-Driven Edit Distance for Trees with Abstract Weights

In this work we define a novel edit distance for trees considered with some abstract weights on the edges. The metric is driven by the idea of considering trees as topological summaries in the context of persistence and topological data analysis. Several examples related to persistent sets are presented. The metric can be computed with a dynamical binary linear programming approach. This framework is applied and further studied in other works focused on merge trees, where the problems of stability and merge trees estimation are also assessed.