  We consider a random process on recursive trees, with three types of events.
Vertices give birth at a constant rate (growth), each edge may be removed
independently (fragmentation of the tree) and clusters (or trees) are frozen
with a rate proportional to their sizes (isolation of connected component). A
phase transition occurs when the isolation is able to stop the growth
fragmentation process and cause extinction. When the process survives, the
number of clusters increases exponentially and we prove that the normalized
empirical measure of clusters a.s. converges to a limit law on recursive trees.
We exploit the branching structure associated with the size of clusters, which
is inherited from the splitting property of random recursive trees. This work
is motivated by the control of epidemics and contact tracing where clusters
correspond to trees of infected individuals that can be identified and
isolated. We complement this work by providing results on the Malthusian
exponent to describe the effect of control policies on epidemics.
