  Assume that the edges of the complete graph $K_n$ are given independent
uniform $[0,1]$ edges weights. We consider the expected minimum total weight
$\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show
that $\mu_k\approx k^2$. Most of the paper is concerned with the case $k=2$. We
show that $\m_2$ tends to an explicitly defined constant and that $\mu_2\approx
4.1704288\ldots$.
