Let _K be the worst-case (supremum) ratio of the weight of the minimum degree-K spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that ₂ = 2 and ₅ = 1. In STOC 94, Khuller, Raghavachari, and Young established the following inequalities: 1.103 < ₃ \le 1.5 and 1.035 < ₄ \le 1.25. We present the first improved upper bounds: ₃ < 1.402 and ₄ < 1.143. As a result, we obtain better approximation algorithms for Euclidean minimum bounded-degree spanning trees. Let _K ^(d) be the analogous ratio in d-dimensional space. Khuller et al. showed that₃ ^(d) < 1.667 for any d. We observe that ₃ ^(d) < 1.633.