Spanning trees of bounded degree

Dirac's classic theorem asserts that if G is a graph on n vertices, and δ(G) ≥ n/2, then G has a hamilton cycle. As is well known, the proof also shows that if deg(x) + deg(y) ≥ (n - 1), for every pair x, y of independent vertices in G, then G has a hamilton path. More generally, S. Win has shown that if k ≥ 2, G is connected and ∑_x∈I deg(x) ≥ n - 1 whenever I is a k-element independent set, then G has a spanning tree T with Δ(T) ≤ k. Here we are interested in the structure of spanning trees under the additional assumption that G does not have a spanning tree with maximum degree less than k. We show that apart from a single exceptional class of graphs, if ∑_x∈I deg(x) ≥ n - 1 for every k-element independent set, then G has a spanning caterpillar T with maximum degree k. Furthermore, given a maximum path P in G, we may require that P is the spine of T and that the set of all vertices whose degree in T is 3 or larger is independent in T.