  A subgraph of an edge-coloured graph is called rainbow if all its edges have
distinct colours. Our main result implies that, given any optimal colouring of
a sufficiently large complete graph $K_{2n}$, there exists a decomposition of
$K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of
Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large
graphs.
