  We study the category whose objects are trees (with or without roots) and
whose morphisms are contractions. We show that the corresponding contravariant
module categories are Noetherian, and we study two natural families of modules
over these categories. The first takes a tree to a graded piece of the homology
of its unordered configuration space, or to the homology of the unordered
configuration space of its cone. The second takes a tree to a graded piece of
the intersection homology of the reciprocal plane of its cone, which is a
vector space whose dimension is given by a Kazhdan-Lusztig coefficient. We
prove finite generation results for each of these modules, which allow us to
obtain results about the growth of Betti numbers of configuration spaces and of
Kazhdan-Lusztig coefficients of graphical matroids.
