  Edit distances between merge trees of scalar fields have many applications in
scientific visualization, such as ensemble analysis, feature tracking or
symmetry detection. In this paper, we propose branch mappings, a novel approach
to the construction of edit mappings for merge trees. Classic edit mappings
match nodes or edges of two trees onto each other, and therefore have to either
rely on branch decompositions of both trees or have to use auxiliary node
properties to determine a matching. In contrast, branch mappings employ branch
properties instead of node similarity information, and are independent of
predetermined branch decompositions. Especially for topological features, which
are typically based on branch properties, this allows a more intuitive distance
measure which is also less susceptible to instabilities from small-scale
perturbations. We describe a quartic runtime algorithm for computing optimal
branch mappings, which is faster than the only other branch
decomposition-independent method in the literature by more than a linear
factor. Furthermore, we compare the results of our method on synthetic and
real-world examples to demonstrate its practicality and utility.
