Summary: Let \(G\) be a connected graph with vertex set \(V (G)\), we define \[\sigma_2 (G)=\min \{d(u)+d(v) \text{ for all non-adjacent vertices } u, v \in V (G)\}.\] Let \(K_{m, m+k}\) be a complete bipartite graph with bipartition \(V (K_{m, m+k}) = A \cup B\), \(|A| = m\), \(|B| = m + k\). Denote by \(H\) the graph obtained from \(K_{m, m+k}\) by adding (or not adding) some edges with two end vertices in \(A\). Let \(\mathcal{H}\) be the set of all connected graphs \(H\). In this paper, we prove that if \(G\) is a connected graph satisfying \(\sigma_2 (G) \geq |G| - k\) then \(G\) has a spanning \(k\)-ended tree except for the case \(G\) is isomorphic to a graph \(H \in \mathcal{H}\). This result is a generalization of the result by H. Broersma and H. Tuinstra [J. Graph Theory 29, No. 4, 227–237 (1998; Zbl 0919.05017)]. On the other hand, we also give a sufficient condition for a graph to have a few branch vertices as a corollary of our main result.