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?