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It is proved that there are constants c1, c2, and c3 such that for any set S of n points in the unit square and for any minimum-lengths of T of S (1) the sum of squares of the edge lengths of T is bounded by c1 log n, (2) the sum of edge lengths of any subset E of T is bounded by c2|E|1/2, and (3) the number of edges having length t or greater in T is at most c3/t2. The second and third bounds are independent of the number of points in S, as well as their locations. Extensions to dimensions d>2 are also sketched.The presence of the logarithmic term in (1) is engaging because such a term is not needed in the case of the minimum spanning tree and several analogous problems, and, furthermore, we know that there always exists some tour of S (which perhaps does not have minimal length) for which the sum of squared edges is bounded independently of n.

It is proved that the length of the longest possible minimum rectilinear Steiner tree of n points in the unit d-cube is asymptotic to Βdn d-1/d, where Βd. In addition to replicating Chung and Graham's exact determination of Β2 = 1, this generalization yields tight new bounds such as 1 ≤ Β3 4 2.