  Centrality measures are widely used to assign importance to graph-structured
data. Recently, understanding the principles of such measures has attracted a
lot of attention. Given that measures are diverse, this research has usually
focused on classes of centrality measures. In this work, we provide a different
approach by focusing on classes of graphs instead of classes of measures to
understand the underlying principles among various measures. More precisely, we
study the class of trees. We observe that even in \fix{the} case of trees,
there is no consensus on which node should be selected as the most central. To
analyze the behavior of centrality measures on trees, we introduce a property
of \emph{tree rooting} that states a measure selects one or two adjacent nodes
as the most important, and the importance decreases from them in all
directions. This property is satisfied by closeness centrality but violated by
PageRank. We show that, for several centrality measures that root trees, the
comparison of adjacent nodes can be inferred by \emph{potential functions} that
assess the quality of trees. We use these functions to give fundamental
insights on rooting and derive a characterization explaining why some measure
root trees. Moreover, we provide an almost liner-time algorithm to compute the
root of a graph by using potential functions. Finally, using a family of
potential functions, we show that many ways of tree rooting exist with
desirable properties.
