  The search of spanning trees with interesting disjunction properties has led
to the introduction of edge-disjoint spanning trees, independent spanning trees
and more recently completely independent spanning trees. We group together
these notions by defining (i, j)-disjoint spanning trees, where i (j,
respectively) is the number of vertices (edges, respectively) that are shared
by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide
some nuances between the existence of disjoint connected dominating sets and
completely independent spanning trees. We prove that determining if there exist
two (i, j)-disjoint spanning trees in a graph G is NP-complete, for every two
positive integers i and j. Moreover we prove that for square of graphs,
k-connected interval graphs, complete graphs and several grids, there exist (i,
j)-disjoint spanning trees for interesting values of i and j.
