Enumeration of Labelled (k, m)-Trees

Ak-graph is called a (k, m)-tree if it can be obtained from a single edge by consecutively adding edges so that every new edge containsk−mnew vertices while its remainingmvertices are covered by an already existing edge. We prove that there are(e(k−m)+m)!(e(km)−e+1)e−2e!m!((k−m)!)edistinct vertex labelled (k, m)-trees witheedges.