  Percolation for a planar lattice has a single percolation threshold, whereas
percolation for a negatively curved lattice displays two separate thresholds.
The enhanced binary tree (EBT) can be viewed as a prototype model displaying
two separate percolation thresholds. We present an analytic result for the EBT
model which gives two critical percolation threshold probabilities,
$p_{c1}=1/2\sqrt{13}-3/2$ and $p_{c2}=1/2$, and yields a size-scaling exponent
$\Phi =\ln [\frac{p(1+p)}{1-p(1-p)}]/\ln 2$. It is inferred that the two
threshold values give exact upper limits and that $p_{c1}$ is furthermore
exact. In addition, we argue that $p_{c2}$ is also exact. The physics of the
model and the results are described within the midpoint-percolation concept:
Monte Carlo simulations are presented for the number of boundary points which
are reached from the midpoint, and the results are compared to the number of
routes from the midpoint to the boundary given by the analytic solution. These
comparisons provide a more precise physical picture of what happens at the
transitions. Finally, the results are compared to related works, in particular,
the Cayley tree and Monte Carlo results for hyperbolic lattices as well as
earlier results for the EBT model. It disproves a conjecture that the EBT has
an exact relation to the thresholds of its dual lattice.
