Equitable partitions into spanning trees in a graph

In this paper we first prove that if the edge set of an undirected graph is the disjoint union of two of its spanning trees, then for every subset P of edges there exists a spanning tree decomposition that cuts P into two (almost) equal parts. The main result of the paper is a further extension of this claim: If the edge set of a graph is the disjoint union of two of its spanning trees, then for every stable set of vertices of size 3, there exists such a spanning tree decomposition that cuts the stars of these vertices into (almost) equal parts. This result fails for 4 instead of 3. The proofs are elementary.