On subgraph number independence in trees

For finite graphs F and G, let Nr(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. I f g and c?? are families of graphs, it is natural to ask then whether or not the quantities NF(G), FE F, are linearly independent when G is restricted to Q. For example, if P = (K1, &} (where K, denotes the complete graph on n vertices) and 9 is the family of all (finite) trees, then of course NK,(T) NK,(T) = 1 for all TE 9. Slightly less trivially, if * = {S, : iz = I, 2, 3,...) (where S, <denotes the star on n edges) and Q again is the family of all trees, then En”=, (l)nflNs (T) = 1 for all TE B. It is proved that such a linear dependence can never occur% S is linite, no FE F has an isolated point, and B contains all trees. This result has important applications in recent work of L. Lo&z and one of the authors (Graham and Lovasz, to appear).