Lets define edge-cycle in a graph 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 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?