Let Gn be the set of graphs with n vertices and H ⊆ Gn. For each H ∈ H, let m(H) = {mi,j (H)}, where mi,j (H) is the number of edges in H that join a vertex of degree i with a vertex of degree j. A vertex-degree-based (VDB, for short) topological index φ is discriminating over H if non-isomorphic graphs in H have different values of φ. We say that φ is weakly discriminating over H if the following weaker condition is satisfied for every H,H′ ∈ H: φ (H) = φ H′ ⇐⇒ m(H) = m H′ . Let CT n be the set of chemical trees with n vertices. In this paper we show that many of the well-known VDB topological indices are not weakly discriminating over CT n. However, the recently introduced Sombor index is weakly discriminating over CT n. Also, we give conditions under which a VDB topological index φ is weakly discriminating over an arbitrary class H ⊆ Gn.