Given an undirected simple graph G with node set V and edge set E, let f(v), for each node v in V, denote a nonnegative integer value that is lower than or equal to the degree of v in G. An f-dominating set in G is a node subset D such that for each node v in V\D, at least f(v) of its neighbors belong to D. In this paper, we study the polyhedral structure of the polytope defined as the convex hull of all the incidence vectors of f-dominating sets in G and give a complete description for the case of trees. We prove that the corresponding separation problem can be solved in polynomial time. In addition, we present a linear-time algorithm to solve the weighted version of the problem on trees: Given a cost c(v) for each node v in V, find an f-dominating set D in G whose cost, given by the sum of the costs of the nodes in D, is minimum.