Related papers: Protection numbers in simply generated trees and P…
The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the…
The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of…
The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in…
P\'olya trees are rooted, unlabeled trees on $n$ vertices. This paper gives an efficient, new way to generate P\'olya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with…
We study protected nodes in $m$-ary search trees, by putting them in context of generalised P\'olya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search…
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned…
We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives…
This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to…
We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and…
We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between…
Place value numbers, such as the binary or decimal numbers can be represented by the end vertices (leaf or pendant vertices) of rooted symmetrical trees. Numbers that consist of at most a fixed number of digits are represented by vertices…
We introduce an extension of the P\'olya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random…
We study the local limit of the fixed-point forest, a tree structure associated to a simple sorting algorithm on permutations. This local limit can be viewed as an infinite random tree that can be constructed from a Poisson point process…
We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton-Watson trees and simply generated trees,…
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…
We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…
Recently, 2-protected nodes were studied in the context of ordered trees and $k$-trees. These nodes have a distance of at least 2 to each leaf. Here, we study digital search trees, which are binary trees, but with a different probability…