  Let $N=\binom{n}{2}$ and $s\geq 2$. Let
$e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$ be $s$ independent permutations of
the edges $E(K_n)$ of the complete graph $K_n$. A {\em MultiTree} is a set
$I\subseteq [N]$ such that the edge sets $E_{I,j}$ induce spanning trees for
$j=1,2,\ldots,s$. In this paper we study the following question: what is the
smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a
hitting time result for $s=2$ and an $O(n\log n)$ bound for $s\geq 3$.
