Spanning trees for many different numbers of leaves

Let G be a connected graph and L(G) the set of all integers k such that G contains a spanning tree with exactly k leaves. We show that for a connected graph G, the set L(G) is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected n-vertex graph – this includes triangulations – has a spanning tree with at least n/2 + 1 leaves, so by a classic theorem of Whitney and our result, in any plane 4-connected n-vertex triangulation one can find for any integer k which is at least 2 and at most n/2 + 1 a spanning tree with exactly k leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many n such that there is a plane 4-connected n-vertex triangulation containing a spanning tree with 2n/3 leaves, but no spanning tree with more than 2n/3 leaves.