A branch vertex in a tree is a vertex of degree at least three. We prove
that, for all s≥1, every connected graph on n vertices with minimum
degree at least (s+31+o(1))n contains a spanning tree having at most
s branch vertices. Asymptotically, this is best possible and solves, in less
general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek,
which was originally motivated by an optimization problem in the design of
optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat