Hindawi Publishing Corporation
Algebra
Volume 2013, Article ID 543913, 11 pages
http://dx.doi.org/10.1155/2013/543913
Research Article
Construction and Composition of Rooted
Trees via Descent Functions
Marco Abrate,
1
Stefano Barbero,
2
Umberto Cerruti,
2
and Nadir Murru
2
1
Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2
Dipartimento di Matematica, Universit` a di Torino, Via Carlo Alberto 8, 10123 Torino, Italy
Correspondence should be addressed to Nadir Murru; nadir.murru@gmail.com
Received 25 March 2013; Revised 28 June 2013; Accepted 30 June 2013
Academic Editor: Ricardo L. Soto Montero
Copyright © 2013 Marco Abrate et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a con-
struction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way,
we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this
paper in order to obtain important results as the creation of new rational and Pythagorean trees.
1. Introduction
Trees represent one of the most important topics in the com-
binatorial theory. Generating trees whose nodes are elements
of a structure may constitute an important step for studying
the properties of the structure itself and for the structure vis-
ualization.
In this paper, some structural properties of trees will be
dealt in a novel way in order to make easy the tree generation
for sets satisfying specifc requirements.
One of the main subjects of the present work is the so-
called descent functions. Identifying functions of this type
allows to reveal a simple and direct technique to the gener-
ation of trees on sets equipped with a weight function over a
partially ordered set. Furthermore, the introduced technique
yields to a substantial simplifcation of the study of tree levels
and easily and directly fnds the degree of any node.
Moreover, we introduce a composition of trees, which
represents a completely new and very powerful tool for the
generation of trees. Tis operation gives the feasibility of gen-
erating a new tree starting from a given pair of trees associated
with a generating system once we assign a partition of the
system itself. We present some examples to show the power of
this tool. In particular, the composition of trees will be used to
construct new examples of trees of rational numbers (rational
trees) and of trees of primitive Pythagorean triples (Pythag-
orean trees). Tese applications are of special interest because
nowadays only three examples of the frst type [1–4] and two
of the latter are known [5–7]. On the other hand, one can
observe that the used techniques allow to generate an infnite
number of trees of either type, and that the shown applica-
tions are merely representative.
Te techniques of explicit construction of the trees
described here show some features that arise from properties
of the algebraic elements involved, making themselves natu-
ral and rigorous.
2. Tree Construction via Descent Functions
In this section, we present a new and useful approach to
rooted trees by using particular functions that we will call
descent functions. Let us start from the following.
Defnition 1. A weight function over is a function :→
, for any set and , such that
(1) is a partially ordered set, where < is the relation of
strict total order;
(2) ∃ ∈ : () < (), for all ̸ =∈;
(3) there are no infnite sequences (
)
∞
=0
, with
∈,
such that for all , (
) > (
+1
).