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 ...

We present two time-e cient state space algorithms for searching minimax trees. Because they are based on SSS* and Dual*, both dominate Alpha-Beta on a node count basis. Moreover, one of them is always faster in searching random trees, even when the leaf node evaluation time is negligible. The fast execution time is attributed to the recursive nature of our algorithms and to their e cient data ...

Abstract Random recursive hypergraphs (RRHs) grow by adding, at each step, a vertex and an edge formed joining the new to randomly chosen existing edge. The model is parameter-free, several characteristics of emerging admit neat expressions via harmonic numbers, Bernoulli Eulerian Stirling numbers first kind. Natural deformations RRHs give rise fascinating models growing random hypergraphs.

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...

In this paper we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional eq...

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...

Abstract. We study the variety of subtrees lying on the fringe of recursive trees and binary search trees by analyzing the distributional behavior of Xn,k, which counts the number of subtrees of size k in a random tree of size n, with k = k(n) dependent on n. Using analytic methods we can characterize for both tree families the phase change behavior of Xn,k as follows. In the subcritical case, ...

The levels of trees are nodes with the same distance to the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp sign-changes when one level is fixed and the other one is varying. We also propose a new means for deriving an asymptotic estimate for the expected width, which is the numbe...

A weighted recursive tree is an evolving in which vertices are assigned random vertex-weights and new connect to a predecessor with probability proportional its weight. Here, we study the maximum degree near-maximum degrees trees when almost surely bounded their distribution function satisfies mild regularity condition near zero. We able specify higher-order corrections first order growth of es...