Bijections of \(k\)-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.

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.

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.

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.

Discrete Mathematics, 2008

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, noncrossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain statistics.

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

Communications in Advanced Mathematical Sciences, 2021

Mathematical trees are connected graphs without cycles, loops and multiple edges. Various trees such as Cayley trees, plane trees, binary trees, d-ary trees, noncrossing trees among others have been studied extensively. Tree-like structures such as Husimi graphs and cacti are graphs which posses the conditions for trees if, instead of vertices, we consider their blocks. In this paper, we use generating functions and bijections to find formulas for the number of noncrossing Husimi graphs, noncrossing cacti and noncrossing oriented cacti. We extend the work to obtain formulas for the number of bicoloured noncrossing Husimi graphs, bicoloured noncrossing cacti and bicoloured noncrossing oriented cacti. Finally, we enumerate plane Husimi graphs, plane cacti and plane oriented cacti according to number of blocks, block types and leaves.

2013

This paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations.

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.

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.