We study the set of allocation rules generated by component efficiency and weighted component fairness, a generalization of component fairness introduced by Herings et al. (2008). Firstly, if the underlying TU-game is superadditive, this set coincides with the core of a graph-restricted game associated with the forest game. Secondly, among this set, only the random tree solutions (Béal et al., 2010) induce Harsanyi payoff vectors for the associated graph-restricted game. We then obtain a new characterization of the random tree solutions in terms of component efficiency and weighted component fairness.