For a martingale (Xn) converging almost surely to a random variable X , the sequence (Xn − X) is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of Régnier’s martingale for the path length in random binary search trees. Grübel and Kabluchko [2014, preprint, arXiv 1410.0469] gave an alte...

We prove a limit theorem for the total Steiner k-distance of a random b-ary recursive tree with weighted edges. The total Steiner k-distance is the sum of all Steiner k-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formul...

We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width, and almost sure convergence. It is widely applicable to random trees of logarithmic height, including...

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton– Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified an...

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn−1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n, the distance ...

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set {1, . . . , n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recurs...

Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r, and for each vertex v 6= r in G, the two different paths from v to r, one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. A recursive circulant graph G(N, d) has N = cdm vertices labeled from 0 to N − 1, where d > 2, m > 1,...

Different multicast routing protocols construct their distribution trees based on the information obtained from the unicast routing infrastructure. Nevertheless, the design of most of these protocols do not take into account that unicast routes may be asymmetric. Indeed, unicast routes in the Internet are very asymmetric. The effects on the quality of the multicast trees vary according to the r...

We study the quantity distance between node j and node n in a random tree of size n chosen from a family of increasing trees. For those subclass of increasing tree families, which can be constructed via a tree evolution process, we give closed formulæ for the probability distribution, the expectation and the variance. Furthermore we derive a distributional decomposition of the random variable c...

Abstract. Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set {1, . . . , n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular intere...