Measuring the distance between merge trees
Topological Methods in Data Analysis and Visualization III: Theory, Algorithms, and Applications, Mathematics and Visualization, pages 151166, 2014. 

TopoInVis

Abstract
Merge trees represent the topology of scalar functions. To assess the topological similarity of functions, one can compare their merge trees. To do so, one needs a notion of a distance between merge trees, which we define. We provide examples of using our merge tree distance and compare this new measure to other ways used to characterize topological similarity (bottleneck distance for persistence diagrams) and numerical difference (L∞norm of the difference between functions).