A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree

Abstract. Let Xn be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for Xn. The basic observation for its proof is that the probability distributions of {Xn : n = 2, 3, . . .} are recursively defined by Xn d = Xn−Dn + 1, n = 2, 3, . . ., X1 = 0, where Dn is a discrete random variable with P{Dn = k} = 1 k(k+1) n n−1 , k = 1, 2, . . . , n − 1, which is independent of (X2, . . . , Xn). This distributional recursion was not studied previously in the sense of weak convergence.