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We perform Monte-Carlo simulations to study the Bernoulli ($p$) bondpercolation on the enhanced binary tree which belongs to the class ofnonamenable graphs with one end. Our numerical results show that the system hastwo different percolation thresholds $p_{c1}$ and $p_{c2}$. All the points inthe intermediate phase $(p_{c1} < p < p_{c2})$ are critical and there existinfinitely many infinite clusters in the intermediate phase. In this phase thecorresponding fractal exponent continuously increases with $p$ from zero tounity.

Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees