2-Binary trees: Bijections and related issues (original) (raw)

Counting formulas and bijections of nondecreasing 2-noncrossing trees

2024

In this paper, we introduce nondecreasing 2-noncrossing trees and enumerate them according to their number of vertices, root degree, and number of forests. We also introduce nondecreasing 2-noncrossing increasing trees and count them by considering their number of vertices, label of the root, label of the leftmost child of the root, root degree, and forests. We observe that the formulas enumerating the newly introduced trees are generalizations of little and large Schröder numbers. Furthermore, we establish bijections between the sets of nondecreasing 2-noncrossing trees, locally oriented noncrossing trees, labelled complete ternary trees, and 3-Schröder paths.

Bijections for ternary trees and non-crossing trees

Discrete Mathematics, 2002

The number Nn of non-crossing trees of size n satisÿes Nn+1 = Tn where Tn enumerates ternary trees of size n. We construct a new bijection to establish that fact. Since Tn = (1=(2n + 1))( 3n n ), it follows that 3(3n − 1)(3n − 2)Tn−1 = 2n(2n + 1)Tn. We construct two bijections "explaining" this recursion; one of them easily extends to the case of t-ary trees.

Bijections for a class of labeled plane trees

We consider plane trees whose vertices are given labels from the set {1, 2, . . . , k} in such a way that the sum of the labels along any edge is at most k + 1; it turns out that the enumeration of these trees leads to a generalization of the Catalan numbers. We also provide bijections between this class of trees and (k + 1)-ary trees as well as generalized Dyck paths whose step sizes are k (up) and 1 (down) respectively, thereby extending some classic results.

Bijections of \(k\)-plane trees

Open Journal of Discrete Applied Mathematics

A \(k\)-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set \(\{1,2,\ldots,k\}\), the children of all vertices are ordered, and if \((i,j)\) is an edge in the tree, where \(i\) and \(j\) are labels of adjacent vertices in the tree, then \(i+j\leq k+1\). In this paper, we construct bijections between these trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths.

On the enumeration of certain sets of planted plane trees

Journal of Combinatorial Theory, 1975

Using the definition of planted plane trees given by D. A. Klarner ("A correspondence between sets of trees," zndag. Math. 31 (1969), 292-296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of (r + l)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.

Enumeration results for alternating tree families

European Journal of Combinatorics, 2010

We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v 1 , v 2 , v 3 , . . . on every path starting at the root of T satisfy

NON-CROSSING TREES, QUADRANGULAR DISSECTIONS, TERNARY TREES, AND DUALITY PRESERVING BIJECTIONS

Using the theory developed in [1] we define an involutory duality for non-crossing trees and provide a bijection between the set of non-crossing trees with n vertices and quadrangular dissec-tions of a 2n-gon by n − 1 non-crossing diagonals that transforms that duality to reflection across an axis connecting the midpoints of two diametrically opposite sides of the 2n-gon. We also show that this bijection fits well with well known bijections involving the set of ternary trees with n − 1 internal vertices and the set of Flagged Perfectly Chain Decomposed Binary Ditrees. Further by analyzing the natural dihedral group action on the set of quadrangular dissections of a 2n-gon we provide closed formulae for the number of quadrangular dissections up to rotations and up to rotations and reflections, the set of non-crossing trees up to rotations and up to rotations and reflections, the number of self-dual non-crossing trees, and the number of oriented and unori-ented unlabeled self-dual non-crossing trees. With the exception of the formula giving the number of unoriented unlabeled non-crossing trees, these formulae are new. 1. THE BIJECTIONS In [1] we introduced a notion of duality (called mind-body duality) for factorizations in a symmetric group S n , and interpreted it in terms of e-v-graphs (that is graphs with ordered edges and vertices) and pegs (that is graphs properly embedded in surfaces with boundary). In this paper we focus on vertex-labeled trees pegged on a disk, or as they are more commonly known, non-crossing trees. We start by fixing conventions and definitions and recalling some basic facts from [1], and refer the reader there for more details. By a non-crossing tree we mean a labeled tree t pegged in D 2 the 2-dimensional disk endowed with the counterclockwise orientation, and we denote the set of non-crossing trees with n vertices by N n. We assume that the vertices of t form the vertices of a regular n-gon and that the order induced by their labels is compatible with the cyclic order of the boundary circle induced by the orientation of the disk, and to be concrete for each n we fix the vertices of a regular n-gon with a standard labeling and we assume that all non-crossing trees have those vertices and that all edges are embedded as chords of the circle. We emphasize that the orientation of the disk is part of the definition and we denote by N ⊺ n the set of trees with n vertices pegged in D 2 ⊺ , the disk endowed with the clockwise orientation. We assume that the elements of N ⊺ n have the same vertices as the elements of N n but with their labels reflected across the diameter that passes through the vertex labeled 1. For a t ∈ N n we denote by t ⊺ the element of N ⊺ n that has the same underlying vertex labeled tree, see the left and middle of Figure 1 for an example. On the other hand the element of N n that is obtained from t by reflecting the edges of t across the diameter passing through 1 will be denoted ¯ t, in other words ¯ t has an edge (n + 2 − i, n + 2 − j) (addition is taken (mod n)) for every edge (i, j) of t. We will sometimes denote by s : N n

Counting trees using symmetries

Journal of Combinatorial Theory, Series A, 2014

We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some results by Knuth and by Bousquet-Mélou and Chapuy about embedded trees. We also give a new proof of the multivariate Lagrange inversion formula. Our strategy for counting trees is to exploit symmetries of refined enumeration formulas: proving these symmetries is easy, and once the symmetries are proved the formulas follow effortlessly. We also adapt this strategy to recover an enumeration formula of Goulden and Jackson for cacti counted according to their degree distribution.

A Combinatorial Identity Concerning Plane Colored Trees and its Applications

2017

In this note, we obtain a combinatorial identity by counting some colored plane trees. This identity has a plethora of implications. In particular, it solves a bijective problem in Stanley’s collection “Bijective Proof Problems”, and gives a formula for the Narayana polynomials, as well as an equivalent expression for the Harer-Zagier formula enumerating unicellular maps.

Refined enumeration of kkk-plane trees and kkk-noncrossing trees

arXiv (Cornell University), 2022

A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than k + 1. These trees are known to be related to (k + 1)-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to (2k + 1)-ary trees.