$[L_{T_N,1},\dots,L_{T_N,n}]$ from above using the AP bound, and then bound it from below by examining the actual number of solutions. With this, we shall show $[L_{T_N,1},\dots,L_{T_N,n}]$ is $2^n = 2^{N-1}$. This agrees with a well known fact in the study of the Kuramoto model: for tree graphs, the original (non-algebraic) Kuramoto model  could have as many as $2^{N-1}$ real equilibria. This shows that even though it is derived from a complex algebraic formulation, the bound $[L_{T_N,1},\dots,L_{T_N,n}]$ on the number of complex solutions can be attained by just real solutions. That is, the algebraization  of  and the extension to the field of complex numbers does not significantly alter the geometry of the underlying problem. For a vertex $i$ in $T_N$, let $\pi(i)$ be the unique parent vertex of $i$, let $\sigma(i)$ be the set of all descendant nodes of $i$, and let $d(i)$ be the depth of the vertex $i$.

\[lem:tree-transform\] The map $\phi = (\phi_1,\dots,\phi_n) : {({\mathbb{C}}^*)}^n \to {({\mathbb{C}}^*)}^n$ given by $$\phi_i(y_1,\dots,y_n) = y_i \, \prod_{k=1}^{d(i)-1} y_{\pi^k(i)}
        \quad\text{for } i=1,\dots,n$$ is a bijection, and the Jacobian matrix $D\phi$ is nonsingular everywhere.

A tree, by definition, has no cycles, so it is always possible to re-index the vertices such that vertex 