  A tree T is invertible if and only if T has a perfect matching. Godsil
considers an invertible tree T and finds that the inverse of the adjacency
matrix of T has entries in {0, 1, -1} and is the signed adjacency matrix of a
graph which contains T. In this paper, we give a new proof of this theorem,
which gives rise to a partial ordering relation on the class of all invertible
trees on 2n vertices. In particular, we show that given an invertible tree T
whose inverse graph has strictly more edges, we can remove an edge from T and
add another edge to obtain an invertible tree T' whose median eigenvalue is
strictly greater. This extends naturally to a partial ordering. We characterize
the maximal and minimal elements of this poset and explore the implications
about the median eigenvalues of invertible trees.
