  We study Bernoulli bond percolation on a random recursive tree of size $n$
with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity.
The sizes of the percolation clusters are naturally stored in a tree. We prove
convergence in distribution of this tree to the genealogical tree of a
continuous-state branching process in discrete time. As a corollary we obtain
the asymptotic sizes of the largest and next largest percolation clusters,
extending thereby a recent work of Bertoin (2014) which deals with cluster
sizes in the supercritical regime. In a second part, we show that the same
limit tree appears in the study of the tree components which emerge from a
continuous-time destruction of a random recursive tree. We comment on the
connection to our first result on Bernoulli bond percolation.
