The rotation graph of binary trees is Hamiltonian

In this paper we study the rotation transformation on binary trees and consider the properties of binary trees under this operation. The rotation is the universal primitive used to rebalance dynamic binary search trees. New binary search tree algorithms have recently been introduced by Sleator and Tarjan. It has been conjectured that these algorithms are as efficient as any algorithm that dynamically restructures the tree using rotations. We hope that by studying rotations in binary trees we shall gain a better understanding of the nature of binary search trees, which in turn will lead to a proof of this “dynamic optimality conjecture”. We define a graph, RG(n), whose vertex set consists of all binary trees containing n nodes, and which has an edge between two trees if they differ by only one rotation. We shall introduce a new characterization of the structure of RG(n) and use it to demonstrate the existence of a Hamiltonian cycle in the graph. The proof is constructive and can be used to enumerate all binary trees with n nodes in constant time per tree.     

On generalized independent subsets of trees

A 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 ?-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 Height of Increasing Trees

Increasing trees have been introduced by Bergeron, Flajolet, and Salvy [1]. This kind of notion covers several well-know classes of random trees like binary search trees, recursive trees, and plane oriented (or heap ordered) trees. We consider the height of increasing trees and prove for several classes of trees (including the above mentioned ones) that the height satisfies EH _n ~ γlogn (for some constant γ > 0) and Var H _n =  O(1) as n → ∞. The methods used are based on generating functions. This research was supported by the Austrian Science Foundation FWF, project S9604, that is part of the Austrian National Research Network "Analytic Combinatorics and Probabilistic Number Theory".     

A study of random Weyl trees

We study binary search trees constructed from Weyl sequences {nθ}, n≥1, where θ is an irrational and {·} denotes “mod 1.” We explore various properties of the structure of these trees, and relate them to the continued fraction expansion of θ. If H_n is the height of the tree with n nodes when θ is chosen at random and uniformly on [0, 1], then we show that in probability, H_n∼(12/π²)log n log log n. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 271–295, 1998     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees

We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper, we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for | U(T) | and thereby arrive at an efficient method of determining | U(T) |. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to the number n of leaves, for totally-ordered trees | U(T) | grows like O(n³). For some special topology types, we are able to obtain simple closed-form formulae for | U(T) |. Using these results, we find a sharp upper bound on | U(T) | and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees measured using the induced SPR-metric. Received May 18, 2004     

Analysis of an algorithm for priority queue administration

A special kind of priority queue structure, priority trees (p-trees), and an algorithm for building such trees are investigated. The main part of the paper is devoted to a mathematical analysis of the algorithm showing that the expected number of key comparisons to insert a random element into a randomp-tree withn nodes isO((logn)²). Also some practical experiments, comparing different types of priority queue strategies, are presented.     

On Rotations and the Generation of Binary Trees

The rotation graph, G_n, has vertex set consisting of all binary trees with n nodes. Two vertices are connected by an edge if a single rotation will transform one tree into the other. We provide a simpler proof of a result of Lucas that G_n, contains a Hamilton path. Our proof deals directly with the pointer representation of the binary tree. This proof provides the basis of an algorithm for generating all binary trees that can be implemented to run on a pointer machine and to use only constant time between the output of successive trees. Ranking and unranking algorithms are developed for the ordering of binary trees implied by the generation algorithm. These algorithms have time complexity O(n²) (arithmetic operations). We also show strong relationships amongst various representations of binary trees and amongst binary tree generation algorithms that have recently appeared in the literature.     

Faster construction of optimal binary split trees

A binary split tree is a search structure combining features of heaps and binary search trees. Building an optimal binary split tree was originally conjectured to be intractable due to difficulties in applying dynamic programming techniques to the problem. However, two algorithms have recently been published which generate optimal trees in O(n⁵) time, for records with distinct access probabilities. An extension allowing nondistinct access probabilities required exponential time. These algorithms consider a range of values when only a single value is possible. A dynamic programming method for determining the correct value is given, resulting in an algorithm which builds an optimal binary split tree in O(n⁵) time for nondistinct access probabilities and Θ(n⁴) time for distinct access probabilities.     

Achiral plane trees

Harary and Robinson showed that the number a_n of achiral planted plane trees with n points coincides with the number p_n of achiral plane trees with n points, for n ? 2. They posed the problem of finding a natural structural correspondence which explains this coincidence. In the present paper this problem is answered by constructing two-to-one correspondences from certain sets of binary sequences to each of the sets of trees concerned, giving a structural basis for the equation 2a_n = 2p_n. Answers are also supplied to similar correspondence-type problems of Harary and Robinson, concerning planted plane trees, and achiral rooted plane trees. In addition, each of these four types of plane trees are counted with numbers of points and endpoints as the enumeration parameters. The results all show a symmetry with respect to the number of endpoints which is not shared by the set of all plane trees.     

Level of nodes in increasing trees revisited

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ { 1,…,n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j,n → ∞ such that n − j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

On the distribution of binary search trees under the random permutation model

We study the distribution Q on the set B_n of binary search trees over a linearly ordered set of n records under the standard random permutation model. This distribution also arises as the stationary distribution for the move-to-root (MTR) Markov chain taking values in B_n when successive requests are independent and identically distributed with each record equally likely. We identify the minimum and maximum values of the functional Q and the trees achieving those values and argue that Q is a crude measure of the “shape” of the tree. We study the distribution of Q(T) for two choices of distribution for random trees T; uniform over B_n and Q. In the latter case, we obtain a limiting normal distribution for −ln Q(T). © 1996 John Wiley & Sons, Inc.     

Twist–Rotation Transformations of Binary Trees and Arithmetic Expressions

The paper studies the computational complexity and efficient algorithms for the twist–rotation transformations of binary trees, which is equivalent to the transformation of arithmetic expressions over an associative and commutative binary operation. The main results are (1) a full binary tree with n labeled leaves can be transformed into any other in at most 3n log n + 2n twist and rotation operations, (2) deciding the twist–rotation distance between two binary trees is NP-complete, and (3) the twist–rotation transformation can be approximated with ratio 6 log n + 4 in polynomial time for full binary trees with n uniquely labeled leaves.     

A note on optimal multiway split trees

Split trees are suitable data structures for storing records with different access frequencies. Under assumption that the access frequencies are all distinct, Huang has proposed anO(n ⁴ logm) time algorithm to construct an (m+1)-way split tree for a set ofn keys. In this paper, we generalize Huang's algorithm to deal with the case of non-distinct access frequencies. The technique used in the generalized algorithm is a generalization of Hesteret al.'s, where the binary case was considered. The generalized algorithm runs inO(n ⁵ logm) time.     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

On the crossing numbers of Cartesian products with trees

Zip product was recently used in a note establishing the crossing number of the Cartesian product K₁,n □ P_m. In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 287–300, 2007     

The expected value of some functions of the convex hull of a random set of points sampled inR d

This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton ^{(d−1)/(d+1)}, which generalizes Rényi and Sulanke’s asymptotic raten ^(1/3) ford=2 and agrees with Raynaud’s asymptotic raten ^{(d−1)/(d+1)} for the expected number of facets, as it should be, by Bárány’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset ofR ^d .     

On martingale tail sums for the path length in random trees

For a martingale (X_n) converging almost surely to a random variable X, the sequence (X_n– X) is called martingale tail sum. Recently, Neininger (Random Structures Algorithms 46 (2015), 346–361) proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko (in press) gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane‐oriented recursive trees. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 493–508, 2017     

Hausdorff metric on faces of the <Emphasis Type="Italic">n</Emphasis>-cube

The Hausdorff metric on all faces of the unit n-cube (I ⁿ ) is considered. The notion of a cubant is used; it was introduced as an n-digit quaternary code of a k-dimensional face containing the Cartesian product of k frame unit segments and the face translation code within I ⁿ . The cubants form a semigroup with a unit (monoid) with respect to the given operation of multiplication. A calculation of Hausdorff metric based on the generalization of the Hamming metric for binary codes is considered. The supercomputing issues are discussed.