Related papers: A One-to-One Correspondence between Natural Number…
We study the question of whether a given regular language of finite trees can be defined in first-order logic. We develop an algebraic approach to address this question and we use it to derive several necessary and sufficient conditions for…
We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here…
A normal network is uniquely determined by the set of phylogenetic trees that it displays. Given a set $\mathcal{P}$ of rooted binary phylogenetic trees, this paper presents a polynomial-time algorithm that reconstructs the unique binary…
An evolutionary tree is a cascade of bifurcations starting from a single common root, generating a growing set of daughter species as time goes by. Species here is a general denomination for biological species, spoken languages or any other…
Motivation: Millions of genes in the modern species belong to only thousands of `gene families'. A gene family includes instances of the same gene in different species (orthologs) and duplicate genes in the same species (paralogs). Genes…
Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an…
We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string…
A common framework is provided that comprises classical ordinal item response models as the cumulative, sequential and adjacent categories models as well as nominal response models and item response tree models. The taxonomy is based on the…
In evolutionary biology, phylogenetic trees are commonly inferred from a set of characters (partitions) of a collection of biological entities (e.g., species or individuals in a population). Such characters naturally arise from molecular…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
A semiorder is a partially ordered set $P$ with two certain forbidden induced subposets. This paper establishes a bijection between $n$-element semiorders of length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection…
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…
In binary and ordinal regression one can distinguish between a location component and a scaling component. While the former determines the location within the range of the response categories, the scaling indicates variance heterogeneity.…
We develop a purely set-theoretic formalism for binary trees and binary graphs. We define a category of binary automata, and display it as a fibred category over the category of binary graphs. We also relate the notion of binary graphs to…
Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the…
A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a…
A decision tree looks like a simple directed acyclic computational graph, where only the leaf nodes specify the output values and the non-terminals specify their tests or split conditions. From the numerical perspective, we express decision…
Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is…
The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call…
In [arXiv:1006.4939] the enumeration order reducibility is defined on natural numbers. For a c.e. set A, [A] denoted the class of all subsets of natural numbers which are co-order with A. In definition 5 we redefine co-ordering for rational…