Multiple isolation of nodes in recursive trees

Abstract

We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate $\ell$ selected nodes in a size-$n$ random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled $1,2,\dots ,\ell$ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled $n+1–\ell ,n+2–\ell ,\dots ,n$ (thus nodes located at the fringe of the tree), and (iii) isolating $\ell$ nodes in the tree, which are selected at random before starting the edge-removal procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behaviour of the number of cuts to isolate all selected nodes, for $\ell$ fixed and $n\to \mathrm{\infty }$.