Planted Plane Trees Research Articles

Bootstrapping and double-exponential limit laws

Combinatorics We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.

The height of watermelons with wall

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalizes a famous result of de Bruijn et al (1972 Graph Theory and Computing (New York: Academic) pp 15–22) on the average height of planted plane trees, and results by Fulmek (2007 Electron. J. Combin. 14 R64) and Katori et al (2008 J. Stat. Phys. 131 1067–83) on the expected value and higher moments, respectively, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information, we prove a reciprocity relation satisfied by the derivatives of one of Jacobi’s theta functions, which generalizes the well-known reciprocity law for Jacobi’s theta functions.

Analysis of three graph parameters for random trees

AbstractWe consider three basic graph parameters, the node‐independence number, the path node‐covering number, and the size of the kernel, and study their distributional behavior for an important class of random tree models, namely the class of simply generated trees, which contains, e.g., binary trees, rooted labeled trees, and planted plane trees, as special instances. We can show for simply generated tree families that the mean and the variance of each of the three parameters under consideration behave for a randomly chosen tree of size n asymptotically ∼μn and ∼νn, where the constants μ and ν depend on the tree family and the parameter studied. Furthermore we show for all parameters, suitably normalized, convergence in distribution to a Gaussian distributed random variable. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Bijections for 2-plane trees and ternary trees

According to the Fibonacci number which is studied by Prodinger et al., we introduce the 2-plane tree which is a planted plane tree with each of its vertices colored with one of two colors and ▪-free. The similarity of the enumeration between 2-plane trees and ternary trees leads us to build several bijections. Especially, we found a bijection between the set of 2-plane trees of n + 1 vertices with a black root and the set of ternary trees with n internal vertices. We also give a combinatorial proof for a relation between the set of 2-plane trees of n + 1 vertices and the set of ternary trees with n internal vertices.

The height of watermelons with wall - Extended Abstract

We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.

On Some Parameters in Heap Ordered Trees

Heap ordered trees are planted plane trees, labelled in such a way that the labels always increase from the root to a leaf. We study two parameters, assuming that nodes are selected at random: the size of the ancestor tree of these nodes and the smallest subtree generated by these nodes. We compute expectation, variance, and also the Gaussian limit distribution, the latter as an application of Hwang's quasi-power theorem.

THE EXFECTED INDEPENDENT DOMINATION NUMBER OF TWO TYPES OF TREES

We deride formulas for the expected values <TEX>$\mu$</TEX>(n) of the independent domination numbers of a random planted plane tree and a random trivalent tree with n vertices, respectively, and we determine the asymptotic behavior of <TEX>$\mu$</TEX>(n) as n goes to infinity.

Bijective Census and Random Generation of Eulerian Planar Maps with Prescribed Vertex Degrees

Abstract: We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular, the result answers a question of Bender and Canfield and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4-regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arquès of the generating function of planar maps with respect to vertices and faces.

Descendants in heap ordered trees or a triumph of computer algebra

A heap ordered tree with $n$ nodes ("size $n$") is a planted plane tree together with a bijection from the nodes to the set $\{1,\dots,n\}$ which is monotonically increasing when going from the root to the leaves. We consider the number of descendants of the node $j$ in a (random) heap ordered tree of size $n\ge j$. Precise expressions are derived for the probability distribution and all (factorial) moments.

DEPTH AND PATH LENGTH OF HEAP ORDERED TREES

A heap ordered tree of size n is a planted plane tree together with a bijection from the nodes to the set {1,…,n} which is monotonically increasing when going from the root to the leaves. In a recent paper by Chen and Ni, the expectation and the variance of the depth of a random node in a random heap ordered tree of size n was considered. Here, we give additional results concerning level polynomials. From a historical point of view, we trace these trees back to an earlier paper by Prodinger and Urbanek and find formulae that are proved in the paper by Chen and Ni by ad hoc computations already in a classic book by Greene and Knuth. Also, we mention that a paper by Bergeron, Flajolet and Salvy develops a more general theory of increasing trees, based on simply generated families of trees. Furthermore we consider the path length which is a natural concept when dealing with the depth. Expectation and variance are obtained, both explicitly and asymptotically.

On the Maximum of a Simple Random Walk

Let $S_0 = 0$, $S_n = \xi _1 + \xi _2 + \cdots + \xi _n $, $n \geq 1$, be the simple random walk generated by a sequence of independent random variables $\xi _i $, $i = 1,2, \ldots $, such that ${\bf P}\{ {\xi _i = 1} \} = 1 - {\bf P}\{ {\xi _i = - 1} \} = \tfrac{1}{2}$, and let T be the moment of the first return of $S_n $ to the state 0. We find an asymptotic representation for the probability ${\bf P}\{ {\max _{0 < k < T} |S_k | > n|T = 2N} \}$ which is exact (in order), assuming that $n^2 N^{ - 1} \to \infty $, and $nN^{ - 1} \leq a < 1$. The results obtained are used to study the asymptotics of moderate and large deviations of the height of a planted plane tree with N vertices.

The Distribution of the Distance to the Root of the Minimal Subtree Containing All the Vertices of a Given Height

Let $h(s)$ be the generating function of the number of direct descendants in a Galton-Watson branching process, $\mu (t)$ the number of particles in the process at time t, $\nu$ the total number of particles bornn in the process during its evolution, and let $\tau (t)$ be the distance to the nearest mutual ancestor of all the particles existing at time t. Assuming that \[ h'(1) = 1,\qquad 0 < B = h''(1) < \infty , \] and the parameters N, $t \to \infty $ in such a way that $t({B /N})^{1/2} \to \beta \in (0,\infty )$, we find the limit \[ \lim {\textbf{P}}\left\{ t^{ - 1} \tau (t ) \leq a\mid\mu ( t ) > 0,\nu = N \right\} = I_\beta ( a ),\quad 0 < a < 1. \] The result obtained is used to find the limiting (as $N \to \infty $) distribution of the distance to the root of the minimal subtree containing all the vertices of a given height in the case where the tree is chosen at random and equiprobably either from the set of all planted plane trees with N nonrooted vertices or from the set of all labelled rooted trees with N vertices.

Limit theorems for the height of a random planted plane tree

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

The Variance of Level Numbers in Certain Families of Trees

For the families of t-ary trees and planted plane trees, the variance of the level numbers (i.e., the depths of specified endnodes) is determined. We apply different types of techniques, such as singularity analysis (Kirschenhofer's diagonalization method) or a probabilistic approach using weak convergence and uniform integrability. In the case of binary trees, an exact variance formula for finite tree size is established; in the other cases, asymptotic equivalents are derived. Also some results on higher moments are presented.

On generalized independent subsets of trees

AbstractA natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k‐independent subsets”). In this paper we give asymptotic results on the average number of ˆ‐independent subsets for trees of size n, where the trees are taken from a so‐called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.

The average height of the d‐th highest leaf of a planted plane tree

AbstractIt is proved that the dth highest leaf, where all planted plane trees with n nodes are assumed to be equally likely, has an average height \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {\pi n} &amp;#x02010; \frac{1}{2} &amp;#x02010; \frac{2}{3}\sum\nolimits_{s = 1}^{d &amp;#x02010; 1} {\left({2/9} \right)^s \left({\begin{array}{*{20}c} {2s + 1} \\ s \\ \end{array}} \right)} + O\left({n^{&amp;#x02010; 1/2 + \varepsilon}} \right) $\end{document} for all ϵ &gt; 0 and n → ∞. This solves a problem left open in one of our previous papers.

On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is &gt; x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.

On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is &amp;gt; x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.

The Average Height of the Second Highest Leaf of a Planted Plane Tree

The 2-height of a planted plane tree is the distance between the root and the second highest leaf (which can be the same as the height). In this note the following theorem is proved: The average 2-height of a planted plane tree with n nodes, considering all such trees to be equally likely, is ( π n ) 1 / 2 − 7 6 + O ( n − 1 / 2 + ε ) , for ε > 0 and n → ∞

On the average shape of simply generated families of trees

AbstractThis paper deals with the limiting distribution of the level number of the jth leaf of a planted plane tree (where leaves are enumerated from left to right) for the so‐called “simply generated families” introduced by Meir and Moon. The mathematical apparatus is determined by the idea of an asymptotic analysis of a given sequence of numbers by studying the location and nature of the singularities of appropriate generating functions.