For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If and are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F, are linearly independent when G is restricted to . For example, if = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and is the family of all (finite) trees, then of course N_K1(T) − N_K2(T) = 1 for all T. Slightly less trivially, if = {S_n: N = 1, 2, 3,…} (where S_n denotes the star on n edges) and again is the family of all trees, then Σ_n=1^∞(−1)ⁿ⁺¹N_Sn(T)=1 for all T. It is proved that such a linear dependence can never occur if is finite, no F has an isolated point, and contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).