Schwinger, ltd: loop-tree duality in the parametric representation

We derive a variant of the loop-tree duality for Feynman integrals in Schwinger parametric representation. This is achieved by decomposing integration domain into disjoint union cells, one each spanning tree graph under consideration. Each these cells total space fiber bundle with contractible fibers over cube. Loop-tree emerges then as result first domain, integrating along bundle. As byproduct we obtain new proof that moduli graphs homotopy equivalent to its spine. In addition, outline potential application Kontsevich's (co-)homology.