The author considers the vertex weighted Steiner tree problem, an extension of the classical Steiner tree problem, from a polyhedral point of view. Given an undirected graph \(G= (V,E)\), a set \(T\subseteq V\), a cost function \(c\) defined on \(E\) and a cost function \(f\) on \(V\), the requirement is to find a Steiner tree \((U,F)\) minimizing the total cost \(c(F)+ f(U)\). This problem can be seen to be a special case of the \(r\)- tree problem. Here, the requirement is to find a tree \((U,F)\) rooted at a given vertex \(r\), minimizing again the total cost \(c(F)+ f(U)\). The author gives formulations of both problems as integer linear programs. To every \(r\)-tree or Steiner tree \((U,F)\) he associates an incidence vector \((x,y)\) defined by \(x_ e= 1\) if \(e\in F\) and 0 otherwise, and \(y_ i= 1\) if \(i\in U\) and 0 otherwise. Denote by \(S_{rT}\subseteq \{0,1\}^{| E|+ | V|}\) the set of incidence vectors of \(r\)-trees and by \(P_{rT}\) the polytope of vectors feasible for the linear programming relaxation. Similarly, \(S_ g\) and \(P_ E\) stand for the corresponding sets of vectors in the relaxation for the vertex weighted Steiner tree problem. In the first part of the paper, the author presents necessary and sufficient conditions for the inequalities in the relaxation of his integer program formulation to define facets of \(\text{conv}(S_{rT})\). The next section deals with the polyhedral characterization of \(\text{conv}(S_{rT})\) for series-parallel graphs. The main theorem proved is that in this case \(P_{rT}\) equals \(\text{conv}(S_{rT})\). Thus, one obtains a complete description of the polytope by linear inequalities, when the underlying graph is series- parallel. The last part of the paper considers the projection \(P_{ST}\) of \(P_ E\) onto the \(x\) variables for the case of a general graph. The author proves a number of necessary conditions for large classes of inequalities to be facet-defining.