On the enumeration of certain sets of planted 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.

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.

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.

arXiv (Cornell University), 2020

We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.

Advances in Applied Mathematics, 1989

Making a bijection between semilabelled trees and some partitions, we build up a powerful theory for enumeration of trees. Theorems of Cayley, Menon, Clarke, Renyi, Erdelyi-Etherington are among the consequences. The theory of random semilabelled trees turns into the theory of random set partitions. Q 1989 Academic Press. Inc.

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.

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

Discrete Applied Mathematics, 2017

Let T = (T, w) be a weighted finite tree with leaves 1, ..., n. For any I := {i 1 , ..., i k } ⊂ {1, ..., n}, let D I (T ) be the weight of the minimal subtree of T connecting i 1 , ..., i k ; the D I (T ) are called k-weights of T . Given a family of real numbers parametrized by the k-subsets of {1, ..., n}, {D I } I∈( {1,...,n} k ) , we say that a weighted tree T = (T, w) with leaves 1, ..., n realizes the family if D I (T ) = D I for any I. In 2006 Levy, Yoshida and Pachter defined, for any positive-weighted tree T = (T, w) with {1, ..., n} as leaf set and any i, j ∈ {1, ..., n}, the numbers S i,j to be Y ∈( {1,...,n}-{i,j} k-2 ) D i,j,Y (T ); they proved that there exists a positive-weighted tree T ′ = (T ′ , w ′ ) such that D i,j (T ′ ) = S i,j for any i, j ∈ {1, ..., n} and that this new tree is, in some way, similar to the given one. In this paper, by using the S i,j defined by Levy, Yoshida and Pachter, we characterize families of real numbers parametrized by {1,...,n} k that are the families of k-weights of weighted trees with leaf set equal to {1, ...., n} and weights of the internal edges positive.