At the beginning of a paper by McKay and Robinson on enumerating eulerian circuits, the authors state that the number of regular tournaments containing a directed rooted tree $T$ on vertices $v_1,\dots,v_n$ with root $v_n$ coincides with the constant term in the generating function $$\prod_{1\le j<k\le n}(x_j^{-1}x_k+x_jx_k^{-1})\,\prod_{jk\in\textrm{ edges of }T}\frac{x_jx_k^{-1}}{x_j^{-1}x_k+x_jx_k^{-1}}\,.$$ Unlike most everything else in the paper, this statement is made without justification, which makes me think that it's either a well-known result or obvious, i.e., except to me.

Could someone provide a reference or a few words to justify this claim?