  A cluster tree provides a highly-interpretable summary of a density function
by representing the hierarchy of its high-density clusters. It is estimated
using the empirical tree, which is the cluster tree constructed from a density
estimator. This paper addresses the basic question of quantifying our
uncertainty by assessing the statistical significance of topological features
of an empirical cluster tree. We first study a variety of metrics that can be
used to compare different trees, analyze their properties and assess their
suitability for inference. We then propose methods to construct and summarize
confidence sets for the unknown true cluster tree. We introduce a partial
ordering on cluster trees which we use to prune some of the statistically
insignificant features of the empirical tree, yielding interpretable and
parsimonious cluster trees. Finally, we illustrate the proposed methods on a
variety of synthetic examples and furthermore demonstrate their utility in the
analysis of a Graft-versus-Host Disease (GvHD) data set.
