For n ≥ 1, let Tn be a random recursive tree (RRT) on the vertex set [n] = {1, . . . , n}. Let degTn (v) be the degree of vertex v in Tn, that is, the number of children of v in Tn. Devroye and Lu [6] showed that the maximum degree ∆n of Tn satisfies ∆n/⌊log2 n⌋ → 1 almost surely; Goh and Schmutz [7] showed distributional convergence of ∆n − ⌊log2 n⌋ along suitable subsequences. In this work we...

The authors consider the limiting behavior of various branches in a uniform recursive tree with size growing to infinity. The limiting distribution of ζn,m, the number of branches with size m in a uniform recursive tree of order n, converges weakly to a Poisson distribution with parameter 1 m with convergence of all moments. The size of any large branch tends to infinity almost surely.

We present a uniform translation from finite-state transducers to regular Böhm trees presentations. The corresponding Böhm tree represents directly the trace semantics of all finite and infinite behaviours of the given transducer. We consider variations on this basic idea, and generalisations of finite-state transducers suggested by the general formalism of regular Böhm trees. This work suggest...

We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially” the same program. The set of all equivalence classes is the category of all algorithms. In order to explore these ideas, the set of primitive recursive functions is considered. Each primi...

Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size n, rescaled by the factor n−1 lnn, converges in pro...

We consider a recursive procedure for destroying rooted trees and isolating a leaf by removing a random edge and keeping the subtree, which does not contain the original root. For two tree families, the simply generated tree families and increasing tree families, we study here the number of random cuts that are necessary to isolate a leaf. We can show limiting distribution results of this param...

Large deviation principles and related results are given for a class of Markov chains associated to the “leaves” in random recursive trees and preferential attachment random graphs, as well as the “cherries” in Yule trees. In particular, the method of proof, combining analytic and Dupuis-Ellis type path arguments, allows for an explicit computation of the large deviation pressure.

We propose a uniform approach for succinct representation of various families of trees. The method is based on two recursive decomposition of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct data structures and has been shown fruitful in succinct representation of ordinal trees [7, 10] and dynamic binary trees [16, 21]. We take an ...

This paper studies the arrangement problem of uniform trees and shows that the arrangement complexity of a uniform tree is either (1) or ((lg n))(> 0). It also presents a recursive algorithm to compute the optimal complete arrangements for (1) arrangeable balanced uniform trees.

We show that the variance of the profiles (number of nodes at each level) of random binary search trees exhibits asymptotically four phase transitions and a bimodal or “two-humped” behavior, in contrast to the unimodality of the mean value of the profiles. Precise asymptotic approximations are derived. The same types of phenomena also hold for the profiles of random recursive trees.