Spanning trees with at most $5$ leaves and branch vertices in total of $K_{1,5}$-free graphs

In this paper, we prove that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_4(G)\ge n-1$ contains a spanning tree with at most $5$ leaves and branch vertices in total. Moreover, the degree sum condition “${\sigma }_4(G)\ge n-1$” is best possible.