the last point of route on level $l$, and the first point of the route on level $l-1$, respectively.[]{data-label="fig:latt"}](fig2b.eps "fig:"){width="30.00000%"} ![Schematic representations of (a) the binary tree and (b) EBT. Arrows show examples of routes from the surface to the midpoint. In panel (a), there is only one possible route from a surface point to the top, whereas in panel (b), there are several because the tree branches are connected. (c) A horizontal layer with a rightward tagged route. The three black dots denote the first point of the route on level $l$, the last point of route on level $l$, and the first point of the route on level $l-1$, respectively.[]{data-label="fig:latt"}](fig2c.eps "fig:"){width="30.00000%"}

The EBT structure is obtained from the usual binary tree by adding horizontal cross links together with a periodic boundary in the horizontal direction, as shown in Figs. \[fig:poin\](c) and \[fig:latt\](b). As is apparent by comparing the hyperbolic lattice in Fig. \[fig:poin\](a) with the EBT in Fig. \[fig:poin\](c), they have similar structures: the essential shared properties are first that the number of lattice points on the surface, $B$, is a finite fraction of the total number of lattice points, $\Omega$, in the limit $\Omega \rightarrow \infty$, and 