Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation al...

We consider growing random recursive trees in random environments, in which at each step a new vertex is attached (by an edge of random length) to an existing tree vertex according to a probability distribution that assigns the tree vertices masses proportional to their random weights. The main aim of the paper is to study the asymptotic behaviour of the distance from the newly inserted vertex ...

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree vertices masses proportional to their random weights. The main aim of the paper is to study the asymptotic behaviour of the distance from the newly inserted vertex...

In this note, we provide a new characterization of Aldous’ Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.

The metric dimension of a graph G is the minimal size subset R vertices that, upon reporting their distance from distinguished (source) vertex v?, enable unique identification source v? among all possible G. In this paper we show Law Large Numbers (LLN) for some classes trees: critical Galton–Watson trees conditioned to have n, and growing general linear preferential attachment trees. former cl...

We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting gene...

We study a new class of time inhomogeneous Pólya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characteriz...

We study a new class of time inhomogeneous Pólya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characteriz...

We consider sums of functions of fringe subtrees of binary search trees and random recursive trees (of total size n). The use of Stein’s method and certain couplings allow provision of simple proofs showing that in both of these trees, the number of fringe subtrees of size k < n, where k → ∞, can be approximated by a Poisson distribution. Combining these results and another version of Stein’s m...

Trees are defined as connected graphs without cycles, and their properties are basics of graph theory. A tree on n nodes labeled 1, 2,..., n is a recursive tree if the node labeled 1 is distinguished as the root, and for each 2 ≤ k ≤ n, the labels of the nodes in the unique path from the root to the node labeled k form an increasing sequence [6]. Bucket trees are a generalization of the ordinar...