We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous in . In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin .
van den Berg, J., Kiss, D., & Nolin, P. (2012). A percolation process on the binary tree where large finite clusters are frozen. Electronic Communications in Probability, 17(2), 1-11. https://doi.org/10.1214/ECP.v17-1694