First non-zero terms for the Taylor expansion at $1$ of the Conway potential function

The Conway potential function $\nabla_L(t_1,\ldots,t_l)$ of an ordered oriented link $L=L_1\cup L_2\cup\ldots\cup L_l\subset S^3$ is considered. In general, this function is not determined by the linking numbers and the Conway potential functions of the components. However, the first two nonzero terms of the Taylor expansion at $1$ of the function $\nabla_L$ are determined by the linking numbers only. We give the explicit formulas for these terms using summation over trees with $l$ vertices.