For the minimal spanning tree on n independent uniform points
in the d-dimensional unit cube, the proportionate number of points of
degree k is known to converge to a limit $\alpha_{k,d}$ as $n \to
\infty$. We show that $\alpha_{k,d}$ converges to a limit $\alpha_k$ as $d \to
\infty$ for each k. The limit $\alpha_k$ arose in earlier work by
Aldous, as the asymptotic proportionate number of vertices of degree k
in the minimum-weight spanning tree on k vertices, when the edge weights
are taken to be independent, identically distributed random variables. We give
a graphical alternative to Aldous's characterization of the $\alpha_k$.