For a graph $G$, let $t(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree. Further, for a vertex $v\in V(G)$, let $t^v(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree, with the extra condition that the tree must contain $v$. The minimum of $t(G)$ ($t^v(G)$, respectively) over all connected triangle-free graphs $G$ (and vertices $v\in V(G)$) on $n$ vertices is denoted by $t_3(n)$ ($t_3^v(n)$). Clearly, $t^v(G)\le t(G)$ for all $v\in V(G)$. In this note, we solve the extremal problem of maximizing $|G|$ for given $t^v(G)$, given that $G$ is connected and triangle-free. We show that $|G|\le 1+\frac{(t_v(G)-1)t_v(G)}{2}$ and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^v(n)=\lceil {1/2}(1+\sqrt{8n-7})\rceil$, improving a recent result by Fox, Loh and Sudakov.
Let $\T_{n}$ be the set of rooted labeled trees on $\set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $\T_{n,k}$ of $\T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.
We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.
Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures.
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Faà di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results …
We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree size goes to infinity.
In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.
The perturbation expansion of the solution of a fixed point equation or of an ordinary differential equation may be expressed as a power series in the perturbation parameter. The terms in this series are indexed by rooted trees and depend on a parameter in the equation in a way determined by the structure of the tree. Power series of this form may be considered more generally; there are two interesting and useful group structures on these series, corresponding to operations of composition and substitution. The composition operation defines the Butcher group, an infinite dimensional group that …
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this construction is a generalization of the $U$-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted $U$-polynomial.
Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley conjectured that his chromatic symmetric function distinguishes all trees, which has remained unresolved. Recently, Hasebe and Tsujie introduced an analogue of Stanley's function for posets, called the strict order quasisymmetric function, and proved that it distinguishes all rooted trees. In this paper, we devise a procedure to explicitly reconstruct a roo…
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
Seo and Shin showed that the number of rooted trees on $[n+1]$ such that the maximal decreasing subtree with the same root has $k+1$ vertices is equal to the number of functions $f:[n]\to[n]$ such that the image of $f$ contains $[k]$. We give a bijective proof of this theorem.
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of $[n]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with $n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.
In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from $[n+1]$ to $[n]$ and the set of triply rooted trees on $[n]$, which leads to the refined enumeration of functions from $[n+1]$ to $[n]$ with respect to the number of elements in the orbit of $n+1$ and the number of period…
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if $\mathcal T$ and $\mathcal T'$ are rooted binary phylogenetic $X$-trees that infers the same set of rooted triples, then $\mathcal T$ and $\mathcal T'$ are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise roote…
Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely unbalanced subtrees, where unbalancing is measured according to the so-called Colless's index. The size of the biggest unbalanced subtree becomes then a new parameter with respect to which we find several enumerations.
The independence polynomial of a graph $G$ is \[I(G,x)=\sum\limits_{k\ge 0}i_k(G)x^k,\] where $i_k(G)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0(G)=1$). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees. In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu's theorem), caterpillars (Wang and Zhu's theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fi…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on $\{1,2,\ldots,n\}$. Classical formulas for counting various trees such as rooted trees, bipartite trees, tripartite trees, plane trees, $k$-ary plane trees, $k$-edge colored trees follow immediately from our recursive relations.
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.
Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(μ_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $μ_r$ is a constant. As an application, we show that the number of any given pattern in $\mathcal{T}_n$ is also asymptotically normally distributed with mean $\sim μ_M n$ and variance $\sim σ_M n$, where $μ_M, σ_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.
The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colore…
Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\binom{n}{k} \, n^{n-k}$. Here I give a simpler proof of this result.
We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find exact closed formulas for expectations and their asymptotic values. Some of these closed formulas are surprisingly simple.
In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\ell$, (ii) non-first-children of degree $k$ at level $\ell-1$, (iii) leaves having $k-1$ elder siblings at level $\ell$, and (iv) non-leaves of outdegree $k$ at level $\ell-1$. Our results unite and generalize several previous works in the literature.
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other…
Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first common ancestor of $v_i$ and $v_j$. We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of $C(T)$, and show that for $d$-ary trees, a specific valu…
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C ρ^{-n} n^{-3/2} where ρis the radius of convergence of T.
The Perron value $ρ(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the moment $μ(T)$ - emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that $μ(T)$ is "almost" an upper bound for $ρ(T)$ and the ratio $μ(T)/ρ(T)$ is unbounded but at most linear in the order of $T$. To achieve these primary goals, we introduce two new…
A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry which means that a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the…
We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.
Based on Hopf algebra of rooted trees introduced by Connes and Kreimer, we construct a class of linear maps on noncommutative polynomial algebra in two indeterminates, namely rooted tree maps. We also prove that their maps induce a class of relations among multiple zeta values.
Let $\mathcal{D}_0:= \{x + iy \ \vert x, y >0\}$, and let $(L, R)$ be a pair of Möbius transformations corresponding to $\mathrm{SL}_2(\mathbb{N}_0)$ matrices such that $R(\mathcal{D}_0)$ and $L(\mathcal{D}_0)$ are disjoint. Given such a pair (called a left-right pair), we can construct a directed graph $\mathcal{F}(L, R)$ with vertices $\mathcal{D}_0$ and edges $\{(z, R(z))\}_{z \in \mathcal{D}_0} \cup \{(z, L(z))\}_{z \in \mathcal{D}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of $\mathrm{SL}_2(\mathbb{N}_…
A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we descri…
We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth's rotation correspondence between planar binary trees and planar rooted trees. The operadic counterpart of the new bijection is explained. Related to this, the space of planar reduced forests is endowed with a combinatorial Hopf algebra structure. The corresponding structure on the space of planar rooted hyperforests is also described.
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way, we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this paper in order to obtain important results as the creation of new rational and Pythagorean trees.
In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.
Let T be the category whose objects are rooted trees and morphisms are order embeddings preserving the root. We prove that finitely generated representations of T are Noetherian using techniques developed by Sam and Snowden which generalize classical Grobner theory. The proof uses a relative version of Kruskals tree Theorem.
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.
Rooted tree maps assign to an element of the Connes-Kreimer Hopf algebra of rooted trees a linear map on the noncommutative polynomial algebra in two letters. Evaluated at any admissible word these maps induce linear relations between multiple zeta values. In this note we show that the derivation relations for multiple zeta values are contained in this class of linear relations.
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
