The relation between the s component Potts model and percolation problems is investigated. The s = 1 limit corresponds to the usual percolation problem (Fortuin and Kasteleyn). It is shown that the s = 0 limit corresponds to a different percolation problem in which the connected clusters have the form of trees. Critical exponents for the Potts model are obtained near six and near one dimension to leading order in 6-d and d-1.