A merge tree (MT) is a topological summary encoding the merging pattern of path-connected components along a filtration of topological spaces. Such objects arise naturally in different scientific fields and fit very well into the framework of Topological Data Analysis. Compared to persistence diagrams (PDs), MTs provide a finer representation of such filtrations, making them an ideal alternative to PDs in some situations. On the other hand, MTs are computationally very expensive and this limits their use in applications. In this talk I will present a metric for MTs which I defined and which can be computed using linear integer programming. This metric satisfies some stability properties (analogous to the ones of 1-Wasserstein metric for PDs), allowing also for some statistic, geometric and topological investigations concerning merge trees estimators and Frechét means. To support the practical relevance of this framework, I will present an application in the field of functional data analysis and one in the field of medical imaging (radiomics).
