This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.
We study random unlabelled $k$-dimensional trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov-Hausdorff-Prokhorov and Benjamini-Schramm limits, that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.
We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov--Hausdorff scaling limit, a Benjamini--Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.
In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or $k$-sided polygons ($k$-gons), where $k\geq 3$ is fixed. This generalization, to $k$-gonal 2-trees, is natural and is closely related, in the planar case, to some specializations of the cell-growth problem. Our goal is the labelled and unlabelled enumeration of $k$-gonal 2-trees according to the number $n$ of $k$-gons. We give explicit formulas in the labelled case, and, in the unlabelled case, recursive and asymptotic formulas.
We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.
We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov--Hausdorff--Prokhorov scaling limit, and converge in a strengthened Benjamini--Schramm sense toward an infinite random graph. We also consider a family of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragm…
This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions assoc…
We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to clog t, where t is the number of vertices, and c=c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k-->oo the height of both processes is asymptotic to log t/(k log 2).
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a compactification of the space of countable (connected) rooted graphs, and use it to generalise the notion of Benjamini-Schramm convergence in order to allow for vertices of infinite degree in the limit object.
Given a permutation $σ$, its corresponding binary search tree is obtained by recursively inserting the values $σ(1),\ldots,σ(n)$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In 1986, Devroye proved that the height of such trees when $σ$ is a random uniform permutation is of order $(c^*+o_\mathbb{P}(1))\log n$ as $n$ tends to infinity, where $c^*$ is the only solution to $c\log(2e/c)=1$ with $c\geq2$. In this paper, we study the height of binary search trees drawn from the record-biased model of…
We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degr…
Random binary search trees are obtained by recursively inserting the elements $σ(1),σ(2),\ldots,σ(n)$ of a uniformly random permutation $σ$ of $[n]=\{1,\dots,n\}$ into a binary search tree data structure. Devroye (1986) proved that the height of such trees is asymptotically of order $c^*\log n$, where $c^*=4.311\ldots$ is the unique solution of $c \log((2e)/c)=1$ with $c \geq 2$. In this paper, we study the structure of binary search trees $T_{n,q}$ built from Mallows permutations. A $\textrm{Mallows}(q)$ permutation is a random permutation of $[n]=\{1,\ldots,n\}$ whose probability is proporti…
Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.
A bijection is constructed between two sets of height restricted lattice paths by means of translating them in two tree classe, namely plane trees and Elena trees. An old bijection between them can be used now for that actual problem.
Plane increasing trees are rooted labeled trees embedded into the plane such that the sequence of labels is increasing on any branch starting at the root. Relaxed binary trees are a subclass of unlabeled directed acyclic graphs. We construct a bijection between these two combinatorial objects and study the therefrom arising connections of certain parameters. Furthermore, we show central limit theorems for two statistics on leaves. We end the study by considering more than 20 subclasses and their bijective counterparts. Many of these subclasses are enumerated by known counting sequences, and th…
A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random variable `sum of distances to the root', defined on any given family of rooted ordered trees (defined by degree restrictions). Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, and expanded by Svante Janson and others, that the limiting (scaled) distributions are all the same, and coincide with the limiting distribution of th…
The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schröder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one corr…
In this tribute to Neil Sloane, we revisit the first sequence in the On-Line Encyclopedia of Integer Sequences, sequence A435 (1, 8, 78, 944, 13800, 237432, 4708144, 105822432, ...), that he encountered when he was a graduate student, and when normalized gives the average total height of rooted labeled trees. We state rigorously-computed explicit expressions for the first twelve moments of the random variable `total height' on rooted labeled trees, and pledge to donate to the OEIS 100 dollars in honor of the first to find an explicit expression for the probability density function of the limit…
We consider various classes of Motzkin trees as well as lambda-terms for which we derive asymptotic enumeration results. These classes are defined through various restrictions concerning the unary nodes or abstractions, respectively: We either bound their number or the allowed levels of nesting. The enumeration is done by means of a generating function approach and singularity analysis. The generating functions are composed of nested square roots and exhibit unexpected phenomena in some of the cases. Furthermore, we present some observations obtained from generating such terms randomly and exp…
We study depth properties of a general class of random recursive trees where each node i attaches to the random node iX_i and X_0, ..., X_n is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (SARRT). We prove that the typical depth D_n, the maximum depth (or height) H_n and the minimum depth M_n of a SARRT are asymptotically given by D_n \sim μ^{-1} \log n, H_n \sim α_{\max} \log n and M_n \sim α_{\min} \log n where μ, α_{\max} and α_{\min} are constants depending only on the distribution of X_0 whenever X_0 has a densit…
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 t…
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 study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.
In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a multiplicative constant. We then apply a natural process "simulate-guess-and-proof" to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height phenomenon (i.e. outside the universal square root behaviour.)
This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height $κ$ holds whenever $κ$ is strongly inaccessible and the coloring takes less than $κ$ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height $κ$, where $κ$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $κ$, whenever $κ$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong…
