Ary Trees Research Articles

Increasing paths in regular trees

We consider a regular $n$-ary tree of height $h$, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of simple paths from the root to a leaf along vertices with increasing labels. We show that if $\alpha = n/h$ is fixed and $\alpha > 1/e$, the probability there exists such a path converges to 1 as $h \to \infty$. This complements a previously known result that the probability converges to 0 if $\alpha \leq 1/e$.

Further Analysis on the Total Number of Subtrees of Trees

When considering the total number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to some other graphical indices in applications. Along this line, it is interesting to study that over some types of trees with a given order, which trees minimize or maximize this number. Here are our main results: (1) The extremal tree which minimizes the total number of subtrees among $n$-vertex trees with $k$ pendants is characterized. (2) The extremal tree which maximizes (resp. minimizes) the total number of subtrees among $n$-vertex trees with a given bipartition is characterized. (3) The extremal tree which minimizes the total number of subtrees among the set of all $q$-ary trees with $n$ non-leaf vertices is identified. (4) The extremal $n$-vertex tree with given domination number maximizing the total number of subtrees is characterized.

An Adaptive Combination Query Tree Protocol for Tag Identification in RFID Systems

A reader must be able to identify tags as quickly as possible, so tag anti-collision is a significant issue for the RFID system. In this paper, we propose a novel tag anti-collision protocol (ACQT), which is suitable for a large mobile tags environment. Based on a 3-ary tree, it has the optimal capacity to identify for tree-based protocols. Using a combination query tree, it can solve the problem of not being able to generate a 3-ary tree when the length of a tag's ID is not a multiple of 3. The joining and leaving strategies can efficiently be applied for tag mobility. The simulation results show that the protocol we present can achieve better performance than previous protocols by decreasing identification delay, collision cycles and idle cycles.

On the size of identifying codes in triangle-free graphs

In an undirected graph G, a subset C⊆V(G) such that C is a dominating set of G, and each vertex in V(G) is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph G, let γID(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected identifiable triangle-free graph G on n vertices having maximum degree Δ≥3, γID(G)≤n−nΔ+o(Δ). This bound is asymptotically tight up to constants due to various classes of graphs including (Δ−1)-ary trees, which are known to have their minimum identifying code of size n−nΔ−1+o(1). We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant c such that the bound γID(G)≤n−nΔ+c holds for any nontrivial connected identifiable graph G.

Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.

On Han's Hook Length Formulas for Trees

Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to $k$-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give combinatorial proofs of Yang's formula for $k$-ary trees and the other formula of Han for binary trees. Our bijections are based on the structure of $k$-ary trees associated with staircase labelings.

A Note on Naturally Embedded Ternary Trees

In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position $j\in\mathbb{Z}$ have positions $j-1$, $j$, and $j+1$. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to $j$, with $j\in\mathbb{N}$. Furthermore, we study the generating function of the number of ternary trees of size $n$ with a given number of internal nodes with label $j$. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank $s$, with $0\le s\le 2n$, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The $2n+1$ external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.

Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators

Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \geq 2$, $k \geq 2$ and $[k] = \{1,\ldots,k\}$. We denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real second-largest (in absolute value) eigenvalue $\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $\sigma_v = \nu_{\xi_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the ``recontruction problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in L_n} \sigma_x. \end{equation*} As noted by Mossel and Peres, when $b\lambda^2 > 1$ (the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$ has uniformly bounded variance. Here, we give bounds on the moment-generating functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have implications for the inference of evolutionary trees.

Abelian state-closed subgroups of automorphisms of $m$-ary trees

The group \mathcal{A}_m of automorphisms of a one-rooted m -ary tree admits a diagonal monomorphism which we denote by x . Let A be an abelian state-closed (or self-similar) subgroup of \mathcal{A}_m . We prove that the combined diagonal and tree-topological closure A^* of A is additively a finitely presented ℤ_m [[ x ]]-module, where ℤ_m is the ring of m -adic integers. Moreover, if A^* is torsion-free then it is a finitely generated pro- m group. Furthermore, the group A splits over its torsion subgroup. We study in detail the case where A^* is additively a cyclic ℤ_m [[ x ]]-module, and we show that when m is a prime number then A^* is conjugate by a tree automorphism to one of two specific types of groups.

A functional limit theorem for the profile of b-ary trees

In this paper we prove a functional limit theorem for the weighted profile of a $b$-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.

Deterministic random walks on regular trees

AbstractJim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order.Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ℤd and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors.This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k‐ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature $\beta_c$, the inverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface area for $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This has been proved for $\Z^2$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for $\beta < \beta_c$ and exponential for $\beta > \beta_c$ were established, where $\beta_c$ is the critical spin-glass parameter, and the tree-height $h$ plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at criticality. The degree of our polynomial bound does not depend on $b$, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for $\beta > \beta_c$, the inverse-gap and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.

Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.

Consistent Minimal Displacement of Branching Random Walks

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be $0$. With $\bar S_v=\max\{S_w: w$ is on the geodesic connecting the root to $v \}$, define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.

The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees

We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.

Infinite labeled trees: From rational to Sturmian trees

This paper studies infinite unordered d -ary trees with nodes labeled by { 0 , 1 } . We introduce the notions of rational and Sturmian trees along with the definitions of (strongly) balanced trees and mechanical trees, and study the relations among them. In particular, we show that (strongly) balanced trees exist and coincide with mechanical trees in the irrational case, providing an effective construction. Such trees also have a minimal factor complexity, hence are Sturmian. We also give several examples illustrating the inclusion relations between these classes of trees.

A note on isoperimetric peaks of complete trees

We prove that the edge isoperimetric peak of the complete k -ary tree of height h is Θ ( h k ) . For the vertex isoperimetric peak we prove a value of Θ ( h ) . Both results improve on the previous best lower bounds of Ω ( h k ) and Ω ( h / k ) , respectively, from [B.V. Subramanya Bharadwaja, L. Sunil Chandran, Bounds on isoperimetric values of trees, Discrete Mathematics 309 (2009) 834–842].

Optimal radio labellings of complete [formula omitted]-ary trees

A radio labelling of a connected graph G is a mapping f : V ( G ) → { 0 , 1 , 2 , … } such that | f ( u ) − f ( v ) | ≥ diam ( G ) − d ( u , v ) + 1 for each pair of distinct vertices u , v ∈ V ( G ) , where diam ( G ) is the diameter of G and d ( u , v ) the distance between u and v . The span of f is defined as max u , v ∈ V ( G ) | f ( u ) − f ( v ) | , and the radio number of G is the minimum span of a radio labelling of G . A complete m -ary tree ( m ≥ 2 ) is a rooted tree such that each vertex of degree greater than one has exactly m children and all degree-one vertices are of equal distance (height) to the root. In this paper we determine the radio number of the complete m -ary tree for any m ≥ 2 with any height and construct explicitly an optimal radio labelling.

Bijections for a class of labeled 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.

Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t) p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system lingers around the critical state for time order h^{-1/2} and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q in (q_t,1) converges, as h tends to 0, to a well-defined measure.