Non-Contiguous Pattern Avoidance in Binary Trees (original) (raw)

Pattern Avoidance in Ternary Trees

This paper considers the enumeration of ternary trees (i.e., rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree patterns; then, for more complex trees, we compute generating functions by extending a known algorithm for pattern-avoiding binary trees. Next, we present an alternate one-dimensional notation for trees which we use to find bijections that explain why certain pairs of tree patterns yield the same avoidance generating function. Finally, we compare our bijections to known "replacement rules" for binary trees and generalize these bijections to a larger class of trees.

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

Patterns in treeshelves

arXiv (Cornell University), 2016

We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.

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.

Pattern avoidance in forests of binary shrubs

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line y = ℓx, for some ℓ ∈ Q + , one of these being the celebrated Duchon's club paths with ℓ = 2/3. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.

Towards a More Precise Analysis of an Algorithm to Generate Binary Trees: A Tutorial

The Computer Journal, 1998

For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of numbers is essential. In the original paper, it was expressed by a recursion. Here, we show how to solve this (and similar) recursions, both explicitly and asymptotically. Some additional information about useful mathematical software is also provided.

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.

Asymptotic enumeration of compacted binary trees of bounded right height

Journal of Combinatorial Theory, Series A, 2020

A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf. We solve the asymptotic counting problem for this class as well as a closely related, further simplified class. For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations. Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity.

Enumeration of trees by inversions

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.