monomial in $F_{T_N}(\mathbf{x})$ must be of the form $x_i x_{\pi(i)}^{-1}$ or $x_i^{-1} x_{\pi(i)}$ for some $i \in \{1,\dots,n\}$. With the substitution given by $x_i = \phi_i(y_1,\dots,y_n)$ for $i=1,\dots,n$, as defined in the previous lemma, it is easy to verify that $$x_i x_{\pi(i)}^{-1} =
        \left(
            y_i \, \prod_{k=1}^{d(i)-1} y_{\pi^k(i)}
        \right)
        \left(
            y_{\pi(i)}^{-1} \, \prod_{k=1}^{d(\pi(i))-1} y_{\pi^k(\pi(i))}^{-1}
        \right)
        = y_i.$$ Therefore the set of monomials which appear in $F_{T_N}(\phi(\mathbf{y}))$ is exactly the set $\{1\} \cup \{ y_1,\dots,y_n \} \cup \{ y_1^{-1},\dots,y_n^{-1} \}$. Under the same transformation, the Adjacency Polytope becomes the cross-polytope $$\operatorname{conv}\left( \bigcup_{i=1}^n \operatorname{conv}( \{\pm \mathbf{e}_i\} ) \right),$$ which is a free sum of the $n$ line segments $\operatorname{conv}( \{\pm \mathbf{e}_i\} )$ for $i=1,\dots,n$. By Lemma \[lem:free sum-vol\], the normalized volume of this polytope is the product of the normalized volume of each of the summands. Since each line segment is of length 2, the AP bound is therefore $2^n = 2^{N-1}$.

We now show the AP bound is actually attainable. That is, there exist choices of complex values for $\{a_{ij}'\}$ and $\omega_1,\dots,\omega_n$ in  for which the system has exactly $2^n$ isolated ${\mathbb{C}}^*$-solutions.

\[lem:tree-reduction\] For the tree graph $T_N$ containing $N = n+1$ vertices, the induced algebraic system  is equivalent to the 