second that there are horizontal paths leading all the way around the lattices. Comparing the binary tree and the EBT in Figs. \[fig:latt\](a) and \[fig:latt\](b) shows that the crucial difference is that the distinct branches for the binary tree are horizontally linked together in the EBT.

Analytic Result
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The method used to find solutions for the EBT model can be illustrated using the binary tree. We pick an arbitrary point on the surface and ask how many routes there are which connect this particular point to the midpoint. For the binary tree the solution is trivial: as seen from Fig. \[fig:latt\](a), there is only one possible route. But in order to reach the top, all the links along this route have to be in place. The chance for a link to be in place is given by the occupation probability $0\leq p\leq
1$. If the number of steps needed to reach the top is $L$, then the probability to get to the top is $P=p^{L},$ which obviously goes to zero for $L\rightarrow \infty$ unless $p=1$. So, if one picks an arbitrary surface point, there is for $p=1$ a finite probability for the existence of a connected route to the 