A tree structure, represented as $\{\infty,3\}$, up to the maximum level $L=10$. (c) The enhanced tree obtained from (b) by connecting the tree branches.[]{data-label="fig:poin"}](fig1b.eps "fig:"){width="20.00000%"} ![Hyperbolic lattice structures drawn on the Poincaré disk. (a) The solid lines show a heptagonal lattice, $\{7,3\}$, and the dotted lines show its dual, $\{3,7\}$. (b) A tree structure, represented as $\{\infty,3\}$, up to the maximum level $L=10$. (c) The enhanced tree obtained from (b) by connecting the tree branches.[]{data-label="fig:poin"}](fig1c.eps "fig:"){width="20.00000%"}

The EBT was introduced in the present context by Nogawa and Hasegawa [@nogawa] as a simplified model for a negatively curved lattice. Like the usual Cayley tree, it retains the key feature that the boundary points are a finite fraction of the total number of lattice points. However, the EBT model has an advantage over the usual hyperbolic lattices because its smaller fraction of boundary points, which makes it easier to obtain numerical results for larger sizes. Even so, recent conflicting numerical results for the EBT model reflect the difficulty inherent in obtaining reliable numerical results for hyperbolic lattices [@com; @reply]. Hence it would be useful to have exact estimates for a prototype model, which can be used as a benchmark for the various 