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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 removedindependently (fragmentation of the tree) and clusters (or trees) are frozenwith a rate proportional to their sizes (isolation of connected component). Aphase transition occurs when the isolation is able to stop the growthfragmentation process and cause extinction. When the process survives, thenumber of clusters increases exponentially and we prove that the normalizedempirical measure of clusters a.s. converges to a limit law on recursive trees.We exploit the branching structure associated with the size of clusters, whichis inherited from the splitting property of random recursive trees. This workis motivated by the control of epidemics and contact tracing where clusterscorrespond to trees of infected individuals that can be identified andisolated. We complement this work by providing results on the Malthusianexponent to describe the effect of control policies on epidemics.

A growth-fragmentation-isolation process on random recursive trees and contact tracing