  In this work a composition-decomposition technique is presented that
correlates tree eigenvectors with certain eigenvectors of an associated
so-called skeleton forest. In particular, the matching properties of a skeleton
determine the multiplicity of the corresponding tree eigenvalue. As an
application a characterization of trees that admit eigenspace bases with
entries only from the set {0, 1,-1} is presented. Moreover, a result due to
Nylen concerned with partitioning eigenvectors of tree pattern matrices is
generalized.
