Related papers: Generating functions for generating trees
The use of machine learning algorithms in finance, medicine, and criminal justice can deeply impact human lives. As a consequence, research into interpretable machine learning has rapidly grown in an attempt to better control and fix…
This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the…
Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{\'e}my showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary…
Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. In…
Using the theoretical basis developed by Yao and Zeilberger, we consider certain graph families whose structure results in a rational generating function for sequences related to spanning tree enumeration. Said families are Powers of Cycles…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
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…
One of the main virtues of trees is to represent formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
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…
The class of self-nested trees presents remarkable compression properties because of the systematic repetition of subtrees in their structure. In this paper, we provide a better combinatorial characterization of this specific family of…
We study the polyregular string-to-string functions, which are certain functions of polynomial output size that can be described using automata and logic. We describe a system of combinators that generates exactly these functions. Unlike…
We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions…
The Boltzmann model for the random generation of "decomposable" combinatorial structures is a set of techniques that allows for efficient random sampling algorithms for a large class of families of discrete objects. The usual requirement of…