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?