numerical techniques (see Refs. [@sykes; @kesten; @wier; @scu; @ziff06; @scu06; @boettcher; @berker] for some existing exact threshold calculations). It is then fortunate that, as shown here, the EBT model itself possesses analytically accessible results.

![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"}](fig2a.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 $l$, 