Assume that the edges of the complete graph Kn
are given independent uniform [0, 1] weights. We consider the expected minimum total weight μk
of k ⩽ 2 edge-disjoint spanning trees. When k is large we show that μk
2. Most of the paper is concerned with the case k = 2. We show that m
2 tends to an explicitly defined constant and that μ
2 ≈ 4.1704288. . . .