The Power of Choice in the Construction of Recursive Trees

The power of choice is known to change the character of random structures and produce desirable optimization effects. We discuss generalizations of random recursive trees, grown under the choice to meet optimization criteria. Specifically, we discuss the random k-minimal (k-maximal) label recursive tree, where a set of k candidate parents, instead of one as in the usual recursive tree, is selected and the node with minimal (maximal) label among them is assigned as parent for the next node. These models are proposed as alternatives for D’Souza et al. (Eur Phys J B59:535–543, 2007) minimal and maximal depth models. The advantage of the label models is that they are tractable and at the same time provide approximations and bounds for the depth models. For the depth of nodes in label models we give the average behavior and exact distributions involving Stirling’s numbers and derive Gaussian limit laws.     

Left and Right Length of Paths in Binary Trees or on a Question of Knuth

We consider extended binary trees and study the joint right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1, . . . , n, and the joint right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the joint limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf j in a random size-n binary tree when j ~ ρn with 0 < ρ > 1, as well as the joint limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-n binary tree. This work was supported by the Austrian Science Foundation FWF, grant S9608-N13.     

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.     

Spectra of Large Random Trees

We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees that we call probability fringe convergence, we show that the empirical spectral distributions for many random tree models converge to a deterministic (model-dependent) limit as the number of vertices goes to infinity. Moreover, the masses assigned by the empirical spectral distributions to individual points also converge in distribution to constants. We conclude for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain lower bounds on the mass assigned to zero by the empirical spectral measures via the connection between the number of zero eigenvalues of the adjacency matrix of a tree and the cardinality of a maximal matching on the tree. In particular, we employ a simplified version of an algorithm due to Karp and Sipser to construct maximal matchings and understand their properties. Moreover, we show that the total weight of a weighted matching is asymptotically equivalent to a constant multiple of the number of vertices when the edge weights are independent, identically distributed, nonnegative random variables with finite expected value, thereby significantly extending a result obtained by Aldous and Steele in the special case of uniform random trees. We greatly generalize a celebrated result obtained by Schwenk for the uniform random trees by showing that if any ensemble converges in the probability fringe sense and a very mild further condition holds, then, with probability converging to one, the spectrum of a realization is shared by at least one other (nonisomorphic) tree. For the linear preferential attachment model with parameter a>?1, we show that for any fixed k, the k largest eigenvalues jointly converge in distribution to a nontrivial limit when rescaled by $n^{1/2\gamma_{a}}$ , where ?? _a =a+2 is the Malthusian rate of growth parameter for an associated continuous-time branching process.     

Analysis of three graph parameters for random trees

We 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     

The shape of large Galton‐Watson trees with possibly infinite variance

Let T be a critical or subcritical Galton‐Watson family tree with possibly infinite variance. We are interested in the shape of T conditioned to have a large total number of vertices. For this purpose we study random trees whose conditional distribution given their size is the same as the respective conditional distribution of T. These random family trees have a simple probabilistic structure if decomposed along the lines of descent of a number of distinguished vertices chosen uniformly at random. The shape of the subtrees spanned by the selected vertices and the root depends essentially on the tail of the offspring distribution: While in the finite variance case the subtrees are asymptotically binary, other shapes do persist in the limit if the variance is infinite. In fact, we show that these subtrees are Galton‐Watson trees conditioned on their total number of leaves. The rescaled total size of the trees is shown to have a gamma limit law. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

On Weighted Depths in Random Binary Search Trees

Following the model introduced by Aguech et al. (Probab Eng Inf Sci 21:133–141, 2007), the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyse weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and the weighted Wiener index in a random binary search tree. We establish three regimes of nodes depending on whether the second-order behaviour of their weighted depths follows from fluctuations of the keys on the path, the depth of the nodes or both. Finally, we investigate a random distribution function on the unit interval arising as scaling limit for weighted depths of nodes with at most one child.     

Hypergraphs as a mean of discovering the dependence structure of a discrete multivariate probability distribution

Most everyday reasoning and decision making is based on uncertain premises. The premises or attributes, which we must take into consideration, are random variables, therefore we often have to deal with a high dimensional multivariate random vector. A multivariate random vector can be represented graphically as a Markov network. Usually the structure of the Markov network is unknown. In this paper we construct special type of junction trees, in order to obtain good approximations of the real probability distribution. These junction trees are capable of revealing some of the conditional independences of the network. We have already introduced the concept of the t-cherry junction tree (E. Kovács and T. Szántai in Proceedings of the IFIP/IIASA//GAMM Workshop on Coping with Uncertainty, 2010), based on the t-cherry tree graph structure. This approximation uses only two and three dimensional marginal probability distributions. Now we use k-th order t-cherry trees, also called simplex multitrees to introduce the concept of the k-th order t-cherry junction tree. We prove that the k-th order t-cherry junction tree gives the best approximation among the family of k-width junction trees. Then we give a method which starting from a k-th order t-cherry junction tree constructs a (k+1)-th order t-cherry junction tree which gives at least as good approximation. In the last part we present some numerical results and some possible applications.     

Subtree Transfer Operations and Their Induced Metrics on Evolutionary Trees

. Leaf-labelled trees are widely used to describe evolutionary relationships, particularly in biology. In this setting, extant species label the leaves of the tree, while the internal vertices correspond to ancestral species. Various techniques exist for reconstructing these evolutionary trees from data, and an important problem is to determine how "far apart" two such reconstructed trees are from each other, or indeed from the true historical tree. To investigate this question requires tree metrics, and these can be induced by operations that rearrange trees locally. Here we investigate three such operations: nearest neighbour interchange (NNI), subtree prune and regraft (SPR), and tree bisection and reconnection (TBR). The SPR operation is of particular interest as it can be used to model biological processes such as horizontal gene transfer and recombination. We count the number of unrooted binary trees one SPR from any given unrooted binary tree, as well as providing new upper and lower bounds for the diameter of the adjacency graph of trees under SPR and TBR. We also show that the problem of computing the minimum number of TBR operations required to transform one tree to another can be reduced to a problem whose size is a function just of the distance between the trees (and not of the size of the two trees), and thereby establish that the problem is fixed-parameter tractable.     

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.     

Optimal Partition Trees

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek (Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n ^1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n ^1+ε ) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n ^1−1/d log ^O(1) n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n ^1−1/d ) query time with high probability. Our method has several advantages:

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     

On the Joint Path Length Distribution in Random Binary Trees

During the 10th Seminar on Analysis of Algorithms , MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary tree. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent . This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n ^5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution's tails and moments. We will also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the Wentzel, Kramers, Brillouin (WKB) method.     

Limit laws for local counters in random binary search trees

Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for w-dependent random variables. Examples include: the number of leaves (L_n), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.     

Embeddings and Other Mappings of Rooted Trees Into Complete Trees

Let T ⁿ be the complete binary tree of height n, with root 1 _n as the maximum element. For T a tree, define and . We disprove a conjecture of Kubicki, Lehel and Morayne, which claims that for any fixed n and arbitrary rooted trees T ₁ T ₂. We show that A(n; T) is of the form where l is the number of leaves of T, and each q _j is a polynomial. We provide an algorithm for calculating the two leading terms of q _l for any tree T. We investigate the asymptotic behaviour of the ratio A(n; T)/C(n; T) and give examples of classes of pairs of trees T ₁, T ₂ where it is possible to compare A(n; T ₁)/C(n; T ₁) and A(n; T ₂)/C(n; T ₂). By calculating these ratios for a particular class of pairs of trees, we show that the conjecture fails for these trees, for all sufficiently large n. Kubicki, Lehel and Morayne have proved the conjecture when T ₁, T ₂ are restricted to being binary trees. We also look at embeddings into other complete trees, and we show how the result can be viewed as one of many possible correlation inequalities for embeddings of binary trees. We also show that if we consider strict order-preserving maps of T ₁, T ₂ into T ⁿ (rather than embeddings) then the corresponding correlation inequalities for these maps also generalise to arbitrary trees.     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999     

Computational experiments with a lazy version of a <Emphasis Type="Italic">K</Emphasis> quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time. Two categories of algorithms are known for ranking simple paths according to the transmission time. One is the adaptation of deviation algorithms for ranking shortest simple paths (Pascoal et al. in Comput. Oper. Res. 32(3):509–520, 2005; Rosen et al. in Comput. Oper. Res. 18(6):571–584, 1991), and another is based on ranking shortest simple paths in a sequence of networks with fixed capacity lower bounds (Chen in Inf. Process. Lett. 50:89–92, 1994), and afterwards selecting the K quickest ones. After reviewing the quickest path and the K-quickest simple paths problems we describe a recent algorithm for ranking quickest simple paths (Pascoal et al. in Ann. Oper. Res. 147(1):5–21, 2006). This is a lazy version of Chen’s algorithm, able to interchange the calculation of new simple paths and the output of each k-quickest simple path. Finally, the described algorithm is computationally compared to its former version, as well as to deviation algorithms.      

Improved algorithms for the multicut and multiflow problems in rooted trees

Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n ²)) greedy algorithm to solve the integer multicut and multiflow problems in a rooted tree. (n is the number of nodes of the tree, and K is the number of commodities). Their algorithm is a special case of the greedy type algorithm of Kolen (Location problems on trees and in the rectilinear plane. Ph.D. dissertation, 1982) to solve weighted covering and packing problems defined by general totally balanced (greedy) matrices. In this communication we improve the complexity bound in Costa et al. (Oper. Res. Lett. 31:21–27, 2003) and show that in the case of the integer multicut and multiflow problems in a rooted tree the greedy algorithm of Kolen can be implemented in subquadratic O(K+n+min (K,n)log n) time. The improvement is obtained by identifying additional properties of this model which lead to a subquadratic transformation to greedy form and using more sophisticated data structures.      

Distributions of Runs and Consecutive Systems on Directed Trees

In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.     

And/or trees: A local limit point of view

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say , and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees , the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).