  The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which
explicitly computes the Borel sum of Feynman perturbation series. This LVE
relies in a crucial way on symmetric tree weights which define a measure on the
set of spanning trees of any connected graph. In this paper we generalize this
method by defining new tree weights. They depend on the choice of a partition
of a set of vertices of the graph, and when the partition is non-trivial, they
are no longer symmetric under permutation of vertices. Nevertheless we prove
they have the required positivity property to lead to a convergent LVE; in
fact, we formulate this positivity property precisely for the first time. Our
generalized tree weights are inspired by the Brydges-Battle-Federbush work on
cluster expansions and could be particularly suited to the computation of
connected functions in QFT. Several concrete examples are explicitly given.
