Related papers: A note on hook length formulas for trees
We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yield a new formula for the number of NATs. We also obtain…
We study the problem of enumerating Braess edges for Kemeny's constant in trees. We obtain bounds and asympotic results for the number of Braess edges in some families of trees.
The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One…
Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes.
We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…
The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history…
This note derives asymptotic upper and lower bounds for the number of planted plane trees on $n$ nodes assigned labels from the set $\{1,2,\ldots, k\}$ with the restriction that on any path from the root to a leaf, the labels must strictly…
We study questions inspired by Erd\H os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the…
This paper investigates the relationships between closed and mixed string amplitudes at the tree level in string theory. Through the analytic continuation of complex variables, we establish a factorization of closed string amplitudes into…
A characterization is provided for each natural number except one (1) by means of an ordered pair of elements. The first element is a natural number called the type of the natural number characterized, and the second is a natural number…
This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a…
A loop-augmented forest is a labeled rooted forest with loops on some of its roots. By exploiting an interplay between nilpotent partial functions and labeled rooted forests, we investigate the permutation action of the symmetric group on…
We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yields a new formula for the number of NATs. We also obtain…
We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…
There is a one-to-one correspondence between natural numbers and rooted trees; the number is called the Matula number of the rooted tree. We show how a large number of properties of trees can be obtained directly from the corresponding…
On trees of fixed order, we show a direct relation between Kemeny's constant and Wiener index, and provide a new formula of Kemeny's constant from the relation with a combinatorial interpretation. Moreover, the relation simplifies proofs of…
The derivation trees of a tree adjoining grammar provide a first insight into the sentence semantics, and are thus prime targets for generation systems. We define a formalism, feature-based regular tree grammars, and a translation from…
We analyze the interplay between labeled trees and the ultrametric spaces they present. We provide characterizations of labeled trees that generate separable ultrametric spaces and those that generate locally finite ultrametric spaces. In…
Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These…
We consider the counting problem of the number of \textit{leaf-labeled increasing trees}, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained…