Sizes of the largest clusters for supercritical percolation on random recursive trees

We consider Bernoulli bond-percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p(n)=1-t/\ln n + o(1/\ln n)$ for some $t>0$ fixed. We show that with high probability, the largest cluster has size close to $\e^{-t}n$ whereas the next largest clusters have size of order $n/\ln n$ only and are distributed according to some Poisson random measure.