  This work addresses the intrinsic relationship between trees and networks
(i.e. graphs). A complete (invertible) mapping is presented which allows trees
to be mapped into weighted graphs and then backmapped into the original tree
without loss of information. The extension of this methodology to more general
networks, including unweighted structures, is also discussed and illustrated.
It is shown that the identified duality between trees and graphs underlies
several key concepts and issues of current interest in complex networks,
including comprehensive characterization of trees and community detection. For
instance, additional information about tree structures (e.g. phylogenetic
trees) can be immediately obtained by taking into account several off-the-shelf
network measurements -- such as the clustering coefficient, degree correlations
and betweenness centrality. At the same time, the hierarchical structure of
networks, including the respective communities, becomes clear when the network
is represented in terms of the respective tree. Indeed, the network-tree
mapping described in this work provides a simple and yet effective means of
community detection.
