Spanning trees with a bounded number of leaves in a claw-free graph

For a graph H and an integer K ≥ 2, let σ _k(H) denote the minimum degree sum of k independent Vertices of H. We prove that if a connected claw-free graph G satisfies σ _k+1 (G) ≥ |G| -k, then G has a spanning tree with at most k leaves. We also show that the bound |G| -k is sharp and discuss the maximum degree of the required spanning trees.