Destruction of Recursive Trees

Abstract

We study for the family of recursive trees, two procedures thatdestroy trees by successively removing edges. In both variants, one starts with atree T of size n andchooses one of the n —1 edges atrandom. Removing thisedge costs atoll depending on the size of T,given by the toll function t _n and leads to twosubtrees T’ and T“. Intheone-sidedvariant,the edge-removal procedure will be iterated with thesubtree containingtheroot,whereas in the two-sided variant it will be iterated with both subtrees. For both variants,we study for toll functions t _n = n ^а withа ≥0 thetotalcosts (= sum of the tolls of every step)obtained bycompletely destroying random recursive trees,where we compute for this quantity the asymptotic behaviour of all moments.