  We study configuration space integral formulas for Milnor's homotopy link
invariants, showing that they are in correspondence with certain linear
combinations of trivalent trees. Our proof is essentially a combinatorial
analysis of a certain space of trivalent "homotopy link diagrams" which
corresponds to all finite type homotopy link invariants via configuration space
integrals. An important ingredient is the fact that configuration space
integrals take the shuffle product of diagrams to the product of invariants. We
ultimately deduce a partial recipe for writing explicit integral formulas for
products of Milnor invariants from trivalent forests. We also obtain cohomology
classes in spaces of link maps from the same data.
