do not involve $x_n$ and form a smaller algebraic system induced by a tree graph consisting of $n$ vertices $0,1,\dots,n-1$. By the induction hypothesis, this smaller system can be transformed into the desired form given in  without altering the solution set. By induction, the statement is true for all tree graphs.

Given a tree graph $T_N$ containing $N$ vertices, there exist choices of complex valued weights ${\{a_{ij}'\}}_{(i,j) \in E(T_N)}$ and complex constants $\omega_1,\dots,\omega_n$, such that the induced system $F_{T_N}(\mathbf{x}) = \mathbf{0}$ has exactly $2^{N-1}$ nonsingular isolated ${\mathbb{C}}^*$-solutions.

By Lemma \[lem:tree-reduction\], the induced algebraic system $F_T(\mathbf{x})$ is equivalent to . Under the transformation $\mathbf{x} = \phi(\mathbf{y})$ given in Lemma \[lem:tree-transform\], $F_{T_N}(\phi(\mathbf{y}))$ is $$\omega^*_i -
                                a_{i,\pi(i)}' \,
        ( y_i - y_i^{-1} )
        = 0
        \quad \text{ for } i = 1,\dots,n
        \label{equ:tree-reduced-y}$$ which has the same number of isolated nonsingular solutions in ${({\mathbb{C}}^*)}^n$ as the original system. Concerning ${\mathbb{C}}^*$-solutions, the $i$-th equation in the above system is equivalent to the quadratic equation $$\omega^*_i
                                -
        a_{i,\pi(i)}' \, y_i^2
        +
        a_{i,\pi(i)}'
        = 0$$ which has exactly two ${\mathbb{C}}^*$-solutions for generic choice of coefficients (even if we require $a_{ij}' = a_{ji}'$). Since there are $n$ independent quadratic equations in $y_1,\dots,y_n$ respectively, the generic root count 