  Much effort has gone into understanding the modular nature of complex
networks. Communities, also known as clusters or modules, are typically
considered to be densely interconnected groups of nodes that are only sparsely
connected to other groups in the network. Discovering high quality communities
is a difficult and important problem in a number of areas. The most popular
approach is the objective function known as modularity, used both to discover
communities and to measure their strength. To understand the modular structure
of networks it is then crucial to know how such functions evaluate different
topologies, what features they account for, and what implicit assumptions they
may make. We show that trees and treelike networks can have unexpectedly and
often arbitrarily high values of modularity. This is surprising since trees are
maximally sparse connected graphs and are not typically considered to possess
modular structure, yet the nonlocal null model used by modularity assigns low
probabilities, and thus high significance, to the densities of these sparse
tree communities. We further study the practical performance of popular methods
on model trees and on a genealogical data set and find that the discovered
communities also have very high modularity, often approaching its maximum
value. Statistical tests reveal the communities in trees to be significant, in
contrast with known results for partitions of sparse, random graphs.
