Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), no. 1, Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs G: every r-colouring of the edge set of G yields r pairwise vertex disjoint monochromatic trees that partition the whole vertex set of G. We determine the threshold for this property for two colours.