0 is the root and $\pi(i) < i$ for any $i$. With this convention, we can write $\phi$ as $\phi(\mathbf{y}) = y^A$ where $y=(y_1,\dots,y_n)$, and $A$ is an $n \times n$ upper triangular integer matrix with all diagonal entries being 1. Then $A$ is a unimodular matrix and hence $A^{-1}$ is also a unimodular integer matrix. It is easy to verify that $\psi(\mathbf{x}) = \mathbf{x}^{A^{-1}}$ is an inverse of $\phi$, and therefore they are both bijections. Moreover, since $\det A = 1$, by Lemma \[lem:toric-automorphism\], $D\phi(\mathbf{y})$ is nonsingular for all $\mathbf{y} \in {({\mathbb{C}}^*)}^n$.

Being a bijection, the transformation $\phi$ given in Lemma \[lem:tree-transform\] preserves the solution count of any system of equations. Moreover, since $D\phi$ remains nonsingular on ${({\mathbb{C}}^*)}^n$, $\phi$ also preserves the more subtle local structures at each solution including multiplicities and local dimensions.

For a tree graph $T_N$ consisting of $N$ nodes, the Adjacency Polytope bound of the induced algebraic system  is $2^{N-1}$.

This result agrees with the general analysis from recent studies [@dekker_synchronization_2013; @dorfler_synchronization_2014]. A similar result for the root counting problem for power-flow equations has been developed in [@Guo1990].

Let $F_{T_N}(\mathbf{x}) = F_{T_N}(x_1,\dots,x_n)$ be the algebraic system  induced by the tree graph $T_N$. Then each non-constant 