  In a complete graph $K_n$ with edge weights drawn independently from a
uniform distribution $U(0,1)$ (or alternatively an exponential distribution
$\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum
weight) and let $T_k$ be the MST after deletion of the edges of all previous
trees $T_i$, $i<k$. We show that each tree's weight $w(T_k)$ converges in
probability to a constant $\gamma_k$ with $2k-2\sqrt k <\gamma_k<2k+2\sqrt k$,
and we conjecture that $\gamma_k = 2k-1+o(1)$. The problem is distinct from
that of Frieze and Johansson (2018), finding $k$ MSTs of combined minimum
weight, and for $k=2$ ours has strictly larger cost.
  Our results also hold (and mostly are derived) in a multigraph model where
edge weights for each vertex pair follow a Poisson process; here we
additionally have $\mathbb E(w(T_k)) \to \gamma_k$. Thinking of an edge of
weight $w$ as arriving at time $t=n w$, Kruskal's algorithm defines forests
$F_k(t)$, each initially empty and eventually equal to $T_k$, with each
arriving edge added to the first $F_k(t)$ where it does not create a cycle.
Using tools of inhomogeneous random graphs we obtain structural results
including that $C_1(F_k(t))/n$, the fraction of vertices in the largest
component of $F_k(t)$, converges in probability to a function $\rho_k(t)$,
uniformly for all $t$, and that a giant component appears in $F_k(t)$ at a time
$t=\sigma_k$. We conjecture that the functions $\rho_k$ tend to time
translations of a single function, $\rho_k(2k+x)\to\rho_\infty(x)$ as $k \to
\infty$, uniformly in $x\in \mathbb R$.
  Simulations and numerical computations give estimated values of $\gamma_k$
for small $k$, and support the conjectures just stated.
