  We find upper bounds for the degrees of vertices and Steiner points in
Steiner Minimal Trees in the d-dimensional Banach spaces \ell_p^d independent
of d. This is in contrast to Minimal Spanning Trees, where the maximum degree
of vertices grows exponentially in d (Robins and Salowe, 1995). Our upper
bounds follow from characterizations of singularities of SMT's due to Lawlor
and Morgan (1994), which we extend, and certain \ell_p-inequalities. We derive
a general upper bound of d+1 for the degree of vertices of an SMT in an
arbitrary smooth d-dimensional Banach space; the same upper bound for Steiner
points having been found by Lawlor and Morgan. We obtain a second upper bound
for the degrees of vertices in terms of 1-summing norms.
