Related papers: Assigned rational functions of a rooted tree
The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
The notion of a root functional of a system of polynomials or ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A root functional is a linear functional that is defined on a polynomial…
The notion of a root functional of polynomials is a generalization of the notion of a root for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of a system of polynomials. A…
We propose a tree-based algorithm for classification and regression problems in the context of functional data analysis, which allows to leverage representation learning and multiple splitting rules at the node level, reducing…
In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…
In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to…
Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework…
We consider the group $\mathrm{Aut}(T)$ of isometries of a semi-homogeneous tree $T=T_{q_+,q_-}$ with valencies $q_+ +1$ and $q_- +1$ and its two orbits $V_+$, $V_-$ respectively. We make use of the action of $\mathrm{Aut} (T)$ to equip the…
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…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted…
The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal…
A record of a rooted Cayley tree is a node whose label is the largest along the unique path to the root. In this work, we find elegant functional equations relating the generating functions for records of rooted Cayley trees and for records…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
We introduce a novel interpretable tree based algorithm for prediction in a regression setting. Our motivation is to estimate the unknown regression function from a functional decomposition perspective in which the functional components…
The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian…
The solution of some equations involving functional derivatives is given as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula. Some algebraic properties of these series are…
We give an elementary characterization of rational functions among meromorphic functions in the complex plane.
To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargava's factorials to a natural combinatorial setting suitable for studying questions around…