Lets define edge-cycle in a graph $G$ as a path where the first
and the last node are adjacent. (in contrast with the definition
of cycle where first and last node are the same). An edge-tree
$T$ is a tree with the additional property that doesn't have an
edge-cycle. In a graph we can compute the number of spanning
trees by using the Matrix-Tree theorem. Is there any similar
theorem for the computation of the number of edge-trees of a
graph?