Plane Trees Research Articles

A Refinement of Leaves on Noncrossing Trees

A leaf of a NC-tree is called a left leaf if it has a label less than that of its parent, and it is called a right leaf otherwise. We enumerate NC-trees with given number of left leaves and right leaves. As an application, the combinatorial interpretation of three identities is obtained. By extending a well-known involution on plane trees to NC-trees, we derive that the two parameters, the number of left leaves and the number of non-root proper vertices, have a symmetric joint distribution in the NC-trees. Finally, a bijection between right leaves and NC-forests is also presented.

Connectivity augmentation in planar straight line graphs

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most ⌊ ( 2 n − 2 ) / 3 ⌋ new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most ⌊ n / 2 ⌋ new edges. These bounds are the best possible. However, for every n ≥ 3 , there are planar straight line trees with n vertices that do not have an embedding preserving augmentation to a 2-edge connected planar straight line graph with fewer than 17 33 n − O ( 1 ) new edges.

Civic and other public planting in ancient Greece

The Western tradition of civic planting originates in ancient Greece, and especially in ancient Athens. Civic planting — the planting of the public spaces of a city for the benefit of all the citiz...

The arithmetic of trees

The arithmetic of the natural numbers ℕ can be extended to arithmetic operations on planar binary trees. This gives rise to a noncommutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and investigate prime trees.

A bijection for triangulations, quadrangulations, pentagulations, etc.

A d-angulation is a planar map with faces of degree d. We present for each integer d ⩾ 3 a bijection between the class of d-angulations of girth d (i.e., with no cycle of length less than d) and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection” which extends an earlier construction of the first author. Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations ( d = 3 ) and by Schaeffer for quadrangulations ( d = 4 ). For d ⩾ 5 , both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate p-gonal d-angulations ( d-angulations with a simple boundary of length p) of girth d. We thereby recover bijectively the results of Brown for simple p-gonal triangulations and simple 2 p-gonal quadrangulations and establish new results for d ⩾ 5 . A key ingredient in our proofs is a class of orientations characterizing d-angulations of girth d. Earlier results by Schnyder and by De Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a d-angulation has girth d if and only if the graph obtained by duplicating each edge d − 2 times admits an orientation having indegree d at each inner vertex.

Phenology of trees and urbanization: a comparative study between New York City and Ithaca, New York

The ‘global warming’ effect has been found to influence vegetation phenological processes. Heat island phenomenon associated with urbanized area presents a unique place to investigate its local warming effects. This study compares the date of budburst (DOBB) of street London plane trees (Platanus × acerifolia) between highly urbanized New York City (NYC) and relatively less urbanized Ithaca, New York in 2007 and 2008. It also linked DOBB with land surface temperature and fractional vegetation cover derived from Landsat satellite images. The DOBB in NYC and Ithaca differed significantly as budburst occurred 3 and 4 days earlier in NYC than in Ithaca in 2007 and 2008, respectively. The intensity of the heat island effect and its effect on tree phenology were greater in NYC. Results show that DOBB can be explained by temperature, and findings could be extrapolated to make inferences on the potential impact of global warming on vegetation communities.

L-Algebras, Triplicial-Algebras, within an Equivalence of Categories Motivated by Graphs

In a previous work, we gave a coalgebraic framework of directed graphs equipped with weights (or probability vectors) in terms of (Markov) L-coalgebras. They are K-vector spaces equipped with two co-operations, Δ M , verifying In this article, we study the category of L-algebras (dual of L-coalgebras), prove that the free L-algebra on one generator is constructed over rooted planar symmetric ternary trees with odd numbers of nodes and the L-operad is Koszul. We then introduce triplicial-algebras: vector spaces equipped with three associative operations verifying three entanglement relations. The free triplicial-algebra is computed and turns out to be related to even trees. Via a general structure theorem (à la Cartier–Milnor–Moore) proved in Section 3, the category of L-algebras turns out to be equivalent to a much more structured category called connected coassociative triplicial-bialgebras (coproduct linked to operations via infinitesimal relations), that is the triple of operads (As, Trip, L) is good. Bidirected graphs, related to NAP-algebras (L-commutative algebras), are briefly evoked and postponed to another article.

Determination of joint reaction forces in a symbolic form in rigid multibody systems

This paper presents an algorithm for determination of joint reaction forces in a symbolic form in planar and spatial tree structure rigid multibody systems . The frictionless revolute and prismatic joints are taken into consideration. The algorithm is based on the use of Kane's equations with undetermined multipliers of constraints. The expressions for the reaction forces and the torques of reaction couples in the joints are obtained in a form that does not require matrix inversion and allows an easy and straightforward implementation in programming environments for symbolic computations (like Mathematica, Maple etc.). The application of the proposed algorithm to the problem of determination of static friction forces in locked Coulomb friction joints is indicated. The algorithm has been illustrated by using both a gymnast on a trampoline and a Puma manipulator. ► The frictionless revolute and prismatic joints are considered. ► Joint reactions in tree structure multibody systems are determined in a symbolic form. ► Kane's equations with undetermined multipliers of constraints are used. ► Theoretical results are illustrated through 2 examples.

The CRT is the scaling limit of unordered binary trees

AbstractWe prove that a uniform, rooted unordered binary tree (also known as rooted, binary Pólya tree) with n leaves has the Brownian continuum random tree as its scaling limit for the Gromov‐Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 467–501, 2011

A new combinatorial identity for unicellular maps, via a direct bijective approach

A unicellular map, or one-face map, is a graph embedded in an orientable surface such that its complement is a topological disk. In this paper, we give a new viewpoint to the structure of these objects, by describing a decomposition of any unicellular map into a unicellular map of smaller genus. This gives a new combinatorial identity for the number ϵg(n) of unicellular maps with n edges and genus g. Unlike the Harer–Zagier recurrence formula, this identity is recursive in only one parameter (the genus).Iterating the construction gives an explicit bijection between unicellular maps and plane trees with distinguished vertices, which gives a combinatorial explanation (and proof) of the fact that ϵg(n) is the product of the n-th Catalan number by a polynomial in n. The combinatorial interpretation also gives a new and simple formula for this polynomial. Variants of the problem are considered, like bipartite unicellular maps, or unicellular maps with only cubic vertices.

Distortion of the Poisson Bracket by the Noncommutative Planck Constants

In this paper we introduce a kind of “noncommutative neighbourhood” of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a “distortion” of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.

Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised which rests on one of two asymptotic scenarios. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals. With a test based on Procrustes tangent space coordinates, not involving the shape space’s curvature, neither can be achieved.

Solution of the Roth-Marques-Durian rotational abrasion model

We solve the rotational abrasion model of Roth, Marques, and Durian [Phys. Rev. E 83, 031303 (2011)], a one-dimensional quasilinear partial differential equation resembling the inviscid Burgers equation with the unusual feature of a step function factor as a coefficient. The complexity of the solution is primarily in keeping track of the cases in the piecewise function that results from certain amputation and interpolation processes, so we also extract from it a model of an evolving planar tree graph that tracks the evolution of the coarse features of the contour.

Boltzmann Samplers, Pólya Theory, and Cycle Pointing

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an “unbiased” way so that a structure of size $n$ gives rise to $n$ pointed structures. We extend Polya theory to the corresponding pointing operator and present a random sampling framework based on both the principles of Boltzmann sampling and Polya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated in several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.

Sur une opérade ternaire liée aux treillis de Tamari

We introduce an anticyclic operad V given by a ternary generator and a quadratic relation. We show that it admits a natural basis indexed by planar binary trees. We then relate this construction to the familly of Tamari lattices (Y n ) n≥0 by defining an isomorphism between V(2n+1) and the Grothendieck group of the category modY n . This isomorphism maps the basis of V(2n+1) to the classes of projective modules and sends the anticyclic map of the operad V(2n+1) to the Coxeter transformation of the derived category of modY n . The Koszul duality theory for operads then allows us to compute the characteristic polynomial of the Coxeter transformation by a Legendre transform.

Condensation in Nongeneric Trees

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability $m<1$. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like $(1-m)N$ where $N$ is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by $2\beta -2$ if the weight $w_n$ of a vertex of degree $n$ is asymptotic to $n^{-\beta}$.

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of ‘connectivity’ for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler (2010) [7] which uses generating functions and Lagrange inversion.

A one-parameter deformation of the Farahat–Higman algebra

We show, by introducing an appropriate basis, that a one-parameter family of Hopf algebras introduced by Foissy [L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (1) (2008) 136–162] interpolates between the Faà di Bruno algebra and the Farahat–Higman algebra. Its structure constants in this basis are deformations of the top connection coefficients, for which we obtain analogues of Macdonald’s formulas.

Q-Legendre transformation: partition functions and quantization of the Boltzmann constant

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton–Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown that this requires a non-commutative analogue of the Planck–Boltzmann constants (ℏ and kB) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the Boltzmann constant with an infinite collection of generators of the so-called epoché (bracketing) algebra. The latter is an infinite-dimensional non-commutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of ‘deformations of deformations’ of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.

Influence of environmental pollution on leaf properties of urban plane trees, Platanus orientalis L.

To investigate whether leaves of plane trees (Platanus orientalis) are damaged by traffic pollution, trees from a megacity (Mashhad, Iran) and a rural area were investigated. Soil and air from the urban centre showed enrichment of several toxic elements, but only lead was enriched in leaves. Leaf size and stomata density were lower at the urban site. At the urban site leaf surfaces were heavily loaded by dust particles but the stomata were not occluded; the cuticle was thinner; other anatomical properties were unaffected suggesting that plane trees can cope with traffic exhaust in megacities.