  Merge trees are a type of graph-based topological summary that tracks the
evolution of connected components in the sublevel sets of scalar functions.
They enjoy widespread applications in data analysis and scientific
visualization. In this paper, we consider the problem of comparing two merge
trees via the notion of interleaving distance in the metric space setting. We
investigate various theoretical properties of such a metric. In particular, we
show that the interleaving distance is intrinsic on the space of labeled merge
trees and provide an algorithm to construct metric 1-centers for collections of
labeled merge trees. We further prove that the intrinsic property of the
interleaving distance also holds for the space of unlabeled merge trees. Our
results are a first step toward performing statistics on graph-based
topological summaries.
