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We study configuration space integral formulas for Milnor's homotopy linkinvariants, showing that they are in correspondence with certain linearcombinations of trivalent trees. Our proof is essentially a combinatorialanalysis of a certain space of trivalent "homotopy link diagrams" whichcorresponds to all finite type homotopy link invariants via configuration spaceintegrals. An important ingredient is the fact that configuration spaceintegrals take the shuffle product of diagrams to the product of invariants. Weultimately deduce a partial recipe for writing explicit integral formulas forproducts of Milnor invariants from trivalent forests. We also obtain cohomologyclasses in spaces of link maps from the same data.

Milnor invariants of string links, trivalent trees, and configuration space integrals