  We consider the model of random trees introduced by Devroye (1999), the
so-called random split trees. The model encompasses many important randomized
algorithms and data structures. We then perform supercritical Bernoulli
bond-percolation on those trees and obtain a precise weak limit theorem for the
sizes of the largest clusters. We also show that the approach developed in this
work may be useful for studying percolation on other classes of trees with
logarithmic height, for instance, we study also the case of $d$-regular trees.
