  Physical phenomena in science and engineering are frequently modeled using
scalar fields. In scalar field topology, graph-based topological descriptors
such as merge trees, contour trees, and Reeb graphs are commonly used to
characterize topological changes in the (sub)level sets of scalar fields. One
of the biggest challenges and opportunities to advance topology-based
visualization is to understand and incorporate uncertainty into such
topological descriptors to effectively reason about their underlying data. In
this paper, we study a structural average of a set of labeled merge trees and
use it to encode uncertainty in data. Specifically, we compute a 1-center tree
that minimizes its maximum distance to any other tree in the set under a
well-defined metric called the interleaving distance. We provide heuristic
strategies that compute structural averages of merge trees whose labels do not
fully agree. We further provide an interactive visualization system that
resembles a numerical calculator that takes as input a set of merge trees and
outputs a tree as their structural average. We also highlight structural
similarities between the input and the average and incorporate uncertainty
information for visual exploration. We develop a novel measure of uncertainty,
referred to as consistency, via a metric-space view of the input trees.
Finally, we demonstrate an application of our framework through merge trees
that arise from ensembles of scalar fields. Our work is the first to employ
interleaving distances and consistency to study a global, mathematically
rigorous, structural average of merge trees in the context of uncertainty
visualization.
