  In this paper, we use tree partition schemes and an algebraic expression for
the virial coefficients in terms of the cluster coefficients in order to derve
upper bounds on the virial coefficients and consequently lower bounds on the
radius of convergence of the virial expansion $\mathcal{R}_{\textrm{Vir}}$. The
bound on the radius of convergence in the case of the Penrose partition scheme
is the same as that proposed by Groeneveld and improves the bound achieved by
Lebowitz and Penrose.
