  In this paper we reformulate in a simpler way the combinatoric core of
constructive quantum field theory We define universal rational combinatoric
weights for pairs made of a graph and one of its spanning trees. These weights
are nothing but the percentage of Hepp's sectors in which the tree is leading
the ultraviolet analysis. We explain how they allow to reshuffle the divergent
series formulated in terms of Feynman graphs into convergent series indexed by
the trees that these graphs contain. The Feynman graphs to be used are not the
ordinary ones but those of the intermediate field representation, and the
result of the reshuffling is called the Loop Vertex Expansion.
