Summary: Let \(G\) be a simple connected graph. For a vertex-degree-based topological index \(TI_f(G) = \sum\limits_{uv\in E(G)}f(d_u, d_v)\), where \(f(x, y)\) is a pertinently chosen symmetric real function, the topological index \(RTI_f(G) = \sum\limits_{uv\in E(G)}\frac{1}{f(d_u, d_v)}\) is called the reciprocal index of \(TI_f\). In this paper, for the first Zagreb index (\(f(x, y) = x + y\)), the second Zagreb index (\(f(x, y) = xy\)), and the forgotten index (\(f(x, y) = x^2 + y^2\)), we prove that the star \(S_n\) and the path \(P_n\) achieve the maximum and minimum values of \(TI_f + RTI_f\) among all trees of order \(n\), respectively. In addition, we show that the same conclusion holds for some other vertex-degree-based topological indices.