for  is exactly $2^n = 2^{N-1}$. Consequently the ${\mathbb{C}}^*$-solution count of the original system $F_{T_N}$ can also reach $2^{N-1}$.

\[cor:index-tree\] Given a tree graph $T_N$ containing $N = n+1$ vertices, let $L_{T_N,1},\dots,L_{T_N,n}$ be the subspace of rational functions defined in . Then $$[\, L_{T_N,1}, \dots, L_{T_N,n}\, ] = 2^n = 2^{N-1}$$

By , $[L_{T_N,1},\dots,L_{T_N,n}]$ is trapped in between the ${\mathbb{C}}^*$-solution count of  and its AP bound. We have shown both to be $2^{N-1}$. Therefore we can conclude $[ L_{T_N,1}, \dots, L_{T_N,n}] = 2^{N-1}$.

By carefully keeping track of the transformation of coefficients, it is possible to argue that the “generic solution count” for  and the original system  are actually the same, thereby establishing Corollary \[cor:index-tree\] directly. However, as noted in Remark \[rmk:generic\], the concept of “generic coefficients” is more subtle than it may appear. Therefore, here we prefer the straightforward calculation of the AP bound over such genericity argument.

Cycle Graphs {#sec:cycle}
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In the study of the Kuramoto model, cycle graphs may be considered as basic building blocks as recent works suggests that it is plausible that detailed analysis of the local geometry near equilibria can be done on a cycle-by-cycle basis [@Bronski2015]. In the context of power-flow study, 