This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.
"Almost giant clusters for percolation on large trees with logarithmic heights." J. Appl. Probab. 50 (3) 603 - 611, September 2013. https://doi.org/10.1239/jap/1378401225