  We perform Monte-Carlo simulations to study the Bernoulli ($p$) bond
percolation on the enhanced binary tree which belongs to the class of
nonamenable graphs with one end. Our numerical results show that the system has
two different percolation thresholds $p_{c1}$ and $p_{c2}$. All the points in
the intermediate phase $(p_{c1} < p < p_{c2})$ are critical and there exist
infinitely many infinite clusters in the intermediate phase. In this phase the
corresponding fractal exponent continuously increases with $p$ from zero to
unity.
