Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial _L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of _L of an m-component link L all of whose Milnor numbers μ_i1…ip vanish for pn. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.