At the beginning of a [paper](http://cs.anu.edu.au/~bdm/papers/euler.pdf) 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?



generating function for regular tournaments
The coefficient of $x\_1^{i\_1}\cdots x\_n^{i\_n}$ in 
$$\prod\_{1\le j<k\le n}(x\_j^{-1}x\_k+x\_jx\_k^{-1})$$
is the number of tournaments on vertices 1, 2, ..., $n$ in which the outdegree minus the indegree of vertex $l$ is $i\_l$ for each $l$. This is because each edge $\{j,k\}$ in the complete graph contributes either 1 to vertex $j$ and $-1$ to vertex $k$ or 
1 to vertex $k$ and $-1$ to vertex $j$, corresponding to the factor $x\_j^{-1}x\_k+x\_jx\_k^{-1}$. So the constant term will count regular tournaments (every vertex has the same indegree as outdegree). If we require that the edge $\{j,k\}$ be directed from $j$ to $k$ then we replace the factor $x\_j^{-1}x\_k+x\_jx\_k^{-1}$ with $x\_j x\_k^{-1}$. (The fact that the edges form a directed rooted tree is not necessary.)


