This paper presents the use of self similarity with random variations and probabilistic growth of branches as the basis of generating natural forms of trees. Bifurcation branching geometry has been considered as the fundamental element in the formation of tree structures and Honda’s model was used to generate the recursive branching patterns. In order to simulate the dynamic variations of natur...

We study the convergence rate to normal limit law for the space requirement of random m-ary search trees. While it is known that the random variable is asymptotically normally distributed for 3 ≤ m ≤ 26 and that the limit law does not exist for m > 26, we show that the convergence rate is O(n) for 3 ≤ m ≤ 19 and is O(n), where 4/3 < α < 3/2 is a parameter depending on m for 20 ≤ m ≤ 26. Our app...

Aldous’ spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which re...

In this paper, we study the tightness of solutions for a family of recursive equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions...

We study first passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph this problem can be described in terms of a continuous time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive trees, which are uniform tr...

Recursive partitioning methods from machine learning are being widely applied in many scientific fields such as, e.g., genetics and bioinformatics. The present work is concerned with the two main problems that arise in recursive partitioning, instability and biased variable selection, from a statistical point of view. With respect to the first issue, instability, the entire scope of methods fro...

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached parent chosen randomly proportionally its weight. Under some assumptions on the sequence of weights, first order for height such has been recently established one authors. In this paper, we obtain second and third orders in asymptotic expansion weighted trees, und...

We consider exact enumerations and probabilistic properties of ranked trees when generated under the random coalescent process. Using a new approach, based on generating functions, we derive several statistics such as the exact probability of finding k cherries in a ranked tree of fixed size n. We then extend our method to consider also the number of pitchforks. We find a recursive formula to c...

We present a computational model to reconstruct ancestor trees of animals with sexual reproduction following the theoretical model presented in Phys. Rev. E 90, 022125 (2014). Through a recursive algorithm combined with a random number generator, it is possible to reproduce the number of ancestor for each generation and use it to constraint a maximum number of the following generation. This sim...