  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.
