  We study minimal vertex covers and maximal matchings on trees. We pay special
attention to the corresponding backbones i.e. these vertices that are occupied
and those that are empty in every minimal vertex cover (resp. these egdes that
are occupied and those that are empty in every maximal matching). The key
result in our approach is that for trees, the backbones can be recovered from a
particular tri-coloring which has a simple characterization. We give
applications to the computation of some averages related to the enumeration of
minimal vertex covers and maximal matchings in the random labeled tree
ensemble, both for finite size and in the asymptotic regime.
