We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Zn, the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Zn − E[Zn], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by...

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 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.

One class of random walks with infinite memory, so-called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact...

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached 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 behavior of the mean numbers of outgoing vertices as the number of steps tends to infinity, under the assumption th...

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...

We present two recursive state space search algorithms that are based on SSS* and Dual*. Both dominate Alpha-Beta on a node count basis and one of them is always faster in searching random trees. These results have been derived on arti cial game trees with negligible leaf node evaluation time. For practical applications with more complex leaf evaluation functions we conjecture that our recursiv...

Exponential tail bounds are derived for solutions of max-recursive equations and for maxrecursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of divide and conquer algorithms, in parallel search algorithms or in the height of random tree models. For the proof we de...