  Hyperbolic structures are obtained by tiling a hyperbolic surface with
negative Gaussian curvature. These structures generally exhibit two percolation
transitions: a system-wide connection can be established at a certain
occupation probability $p=p_{c1}$ and there emerges a unique giant cluster at
$p_{c2} > p_{c1}$. There have been debates about locating the upper transition
point of a prototypical hyperbolic structure called the enhanced binary tree
(EBT), which is constructed by adding loops to a binary tree. This work
presents its lower bound as $p_{c2} \gtrsim 0.55$ by using phenomenological
renormalization-group methods and discusses some solvable models related to the
EBT.
