Bijections for Ranked Tree-Child Networks

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.

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.

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.

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.

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.

Discrete Mathematics, 1980

We deal with the class T,, of ordered trees with n edges. Several enumeration problems concerning Tn and some of its combinatorial properties are studied.

Journal of Graph Theory, 1995

Mallows and Riordan 21] rst de ned the inversion polynomial, J n (q), for trees with n vertices and found its generating function. In the present work, we de ne inversion polynomials for ordered, plane and cyclic trees and nd their values at q = 0; 1. Our techniques involve the use of generating functions (including Lagrange inversion), hypergeometric series and binomial coe cient identities, induction and bijections. We also derive asymptotic formulae for those results for which we do not have a closed form.

Discrete Applied Mathematics

Galled trees are widely studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into galled networks by relaxing a structural condition. In this work, a linear recurrence formula is given for counting 1galled networks, which are galled networks satisfying the condition that each reticulate node has only one leaf descendant. Since every galled network consists of a set of 1-galled networks stacked one on top of the other, a method is also presented to count and enumerate galled networks.

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.

2020

There exists a wealth of literature concerning families of increasing trees, particularly suitable for representing the evolution of either data structures in computer science, or probabilistic urns in mathematics, but are also adapted to model evolutionary trees in biology. The classical notion of increasing trees corresponds to labeled trees such that, along paths from the root to any leaf, node labels are strictly increasing; in addition nodes have distinct labels. In this paper we introduce new families of increasingly labeled trees relaxing the constraint of unicity of each label. Such models are especially useful to characterize processes evolving in discrete time whose nodes evolve simultaneously. In particular, we obtain growth processes for biology much more adequate than the previous increasing models. The families of monotonic trees we introduce are much more delicate to deal with, since they are not decomposable in the sense of Analytic Combinatorics. New tools are requi...