Number Of Subtrees Research Articles

Complexity of Two-Variable Logic on Finite Trees

Verification of properties expressed in the two-variable fragment of first-order logic FO 2 has been investigated in a number of contexts. The satisfiability problem for FO 2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO 2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees, FO 2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO 2 gives a 2EXPTIME bound for satisfiability of FO 2 over trees. This work contains a comprehensive analysis of the complexity of FO 2 on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO 2 , its guarded version, GF 2 . Our results depend on an analysis of types in models of FO 2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO 2 sentences over finite trees.

An edge-swap heuristic for finding dense spanning trees

Finding spanning trees under various restrictions has been an interesting question to researchers. A “dense” tree, from a graph theoretical point of view, has small total distances between vertices and large number of substructures. In this note, the “density” of a spanning tree is conveniently measured by the weight of a tree (defined as the sum of products of adjacent vertex degrees). By utilizing established conditions and relations between trees with the minimum total distance or maximum number of subtrees, an edge-swap heuristic for generating “dense” spanning trees is presented. Computational results are presented for randomly generated graphs and specific examples from applications.

Subtrees of spiro and polyphenyl hexagonal chains

The number of subtrees of a graph and its variations are among popular topological indices that have been vigorously studied. We investigate the subtree numbers of spiro and polyphenyl hexagonal chains, molecular graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. We first present the generating functions for subtrees, through which explicit formulas for computing the subtree number of spiro and polyphenyl hexagonal chains are obtained. We then establish a relation between the subtree numbers of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain. This allows us to show that the spiro and polyphenyl hexagonal chains with the minimum (resp. second minimum, third minimum) subtree numbers coincide with the ones that attain the maximum (resp. second maximum, third maximum) Wiener indices, and vice versa. The subtree densities of these hexagonal chains are also briefly discussed.

MEASURING COMPLEXITY OF DOMAIN MODELS REPRESENTED BY FEATURE DIAGRAMS

Feature models represented by Feature Diagrams (FDs) prevail in the software product line approach. The product line approach and FDs are used to manage variability and complexity of software families and to ensure higher quality and productivity of product development through higher-level feature modeling and reuse. In this paper we, first, analyze the properties of feature models. Then, combining some properties of FDs with ideas of Miller’s, Metcalfe’s and Keating’s works, we propose three FD complexity measures. The first measure gives boundaries to estimate cognitive complexity of a generic component to be derived from the feature model. The second measure describes structural complexity of the model expressed through the number of adequate sub-trees of the given model. The third measure estimates total cognitive and structural complexity of the model. To validate the introduced measures, we present a case study with three feature models of a varying complexity.

Extending partial representations of subclasses of chordal graphs

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T′ and some pre-drawn subtrees of T′. It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e., keeps the pre-drawn subtrees unchanged).We consider four modifications of T′ leading to vastly different problems: (i) T=T′, (ii) T is a subdivision of T′, (iii) T is a supergraph of T′, and (iv) T′ is a topological minor of T. In some cases, it is interesting to consider the complexity even when just T′ is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization. We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components of the input graph G and the size of the tree T′.We describe an interesting relation with integer partition problems. The problem 3-Partition is used for all NP-completeness reductions. When the space in T′ is limited, partial representation extension of proper interval graphs is “equivalent” to the BinPacking problem.

A k-d tree-based algorithm to parallelize Kriging interpolation of big spatial data

Parallel computing provides a promising solution to accelerate complicated spatial data processing, which has recently become increasingly computationally intense. Partitioning a big dataset into workload-balanced child data groups remains a challenge, particularly for unevenly distributed spatial data. This study proposed an algorithm based on the k-d tree method to tackle this challenge. The algorithm constructed trees based on the distribution variance of spatial data. The number of final sub-trees, unlike the conventional k-d tree method, is not always a power of two. Furthermore, the number of nodes on the left and right sub-trees is always no more than one to ensure a balanced workload. Experiments show that our algorithm is able to partition big datasets efficiently and evenly into equally sized child data groups. Speed-up ratios show that parallel interpolation can save up to 70% of the execution time of the consequential interpolation. A high efficiency of parallel computing was achieved when the datasets were divided into an optimal number of child data groups.

Limit laws for functions of fringe trees for binary search trees and random recursive trees

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings. As a consequence, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree or in the random recursive tree (of total size $ n $) has an asymptotical Poisson distribution if $ k\rightarrow\infty $, and that the distribution is asymptotically normal for $ k=o(\sqrt{n}) $. Furthermore, we prove similar results for the number of subtrees of size $ k $ with some required property $ P $, e.g., the number of copies of a certain fixed subtree $ T $. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. <br /><br />We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of $ \ell $-protected nodes in a binary search tree or in a random recursive tree.

The distances between internal vertices and leaves of a tree

Distance-based graph invariants have been of great interest and extensively studied. The classic Wiener index was proposed in biochemistry and defined to be the sum of distances between all pairs of vertices. The sum of distances between all pairs of leaves, named the Gamma index and the terminal Wiener index respectively, was motivated from both biochemistry and phylogenetic reconstruction. The studies of such distance-based graph invariants include, for instance, the “middle part” of a tree, the extremal structures with given constraints, the extremal values of ratios of the distance function at the “middle part” and leaves. In particular, when considering the extremal structures, correlations between the Wiener index and other graph invariants have been discovered and general methods have been developed. Other related graph invariants include the number of subtrees or leaf containing subtrees (corresponding to the “acceptable residue configurations”), subforests, root-containing subtrees (in relation to the antichains in a Hasse diagram with the structure of a rooted tree), to name a few.As has been repeatedly mentioned in earlier works, it is of both mathematical interests and practical importance to generalize the current studies to other distance-based graph invariants. The sum of distances between internal vertices has been considered and many similar results have been obtained.On the other hand, a natural similar concept is the sum of distances between internal vertices and leaves. It has been proposed in different literatures. In this paper we conduct a relatively comprehensive study of this concept, providing results analogous to those of the aforementioned studies. We start with identifying the “middle part” of a tree with respect to the total distance from leaves. Then the extremal ratios with respect to the distance from leaves are examined and the structures achieving the extremal ratios are characterized. Lastly, we provide extremal trees under different constraints that maximize or minimize the sum of all such distances.

Coding multitype forests: Application to the law of the total population of branching forests

By extending the breadth first search algorithm to any $d$-type critical or subcritical irreducible branching tree, we show that such trees may be encoded through $d$ independent, integer valued, $d$-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process.

Extremal values of ratios: Distance problems vs. subtree problems in trees II

We discovered a dual behavior of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems (Székely and Wang, 2006, 2005). We introduced the concept of subtree core: the subtree core of a tree consists of one or two adjacent vertices of a tree that are contained in the largest number of subtrees. Let σ(T) denote the sum of distances between unordered pairs of vertices in a tree T and σT(v) the sum of distances from a vertex v to all other vertices in T. Barefoot et al. (1997) determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. Let F(T) denote the number of subtrees of T and FT(v) the number of subtrees containing v in T. In Part I of this paper we tested how far the negative correlation between distances and subtrees go if we look for (and characterize) the extremal values of FT(w)/FT(u), FT(w)/FT(v). In this paper we characterize the extremal values of F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and w is a leaf in T-completing the analogy, changing distances to the number of subtrees.

Robust node-disjoint multipath routing for wireless sensor networks

In wireless sensor networks WSNs, disjoint multipath routing has gotten immense research interest due to its capability of providing high reliability and load balancing. In this paper, we present a robust node-disjoint multipath routing RNMR protocol for WSNs. The protocol first constructs a multi-routing tree which has a number of node-joint but link-disjoint subtrees, and then finds multiple paths along these subtrees. Furthermore, by taking advantage of the structural characteristics inherent in the multi-routing tree, we present a customised redundancy scheme to improve the reliability of data transmission. The scheme provides individualised data redundancy for each sensor node. The protocol does not need to employ any control beacons in the data routing phase. Theoretical analysis and simulation results show that our protocol achieves higher reliability, reduces end-to-end delay and improves energy efficiency compared with existing approaches.

The Minimal Number of Subtrees with a Given Degree Sequence

In this paper, we investigate the structures of extremal trees which have the minimal number of subtrees in the set of all trees with a given degree sequence. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $${\pi = (d_1, \ldots, d_{5}, 1, \ldots, 1)}$$ ? = ( d 1 , ? , d 5 , 1 , ? , 1 ) have been characterized.

The Minimal Number of Subtrees of a Tree

In this note, we consider the trees (caterpillars) that minimize the number of subtrees among trees with a given degree sequence. This is a question naturally related to the extremal structures of some distance based graph invariants. We first confirm the expected fact that the number of subtrees is minimized by some caterpillar. As with other graph invariants, the specific optimal caterpillar is nearly impossible to characterize and depends on the degree sequence. We provide some simple properties of such caterpillars as well as observations that will help finding the optimal caterpillar.

Yule-generated trees constrained by node imbalance

The Yule process generates a class of binary trees which is fundamental to population genetic models and other applications in evolutionary biology. In this paper, we introduce a family of sub-classes of ranked trees, called Ω-trees, which are characterized by imbalance of internal nodes. The degree of imbalance is defined by an integer 0≤ω. For caterpillars, the extreme case of unbalanced trees, ω=0. Under models of neutral evolution, for instance the Yule model, trees with small ω are unlikely to occur by chance. Indeed, imbalance can be a signature of permanent selection pressure, such as observable in the genealogies of certain pathogens. From a mathematical point of view it is interesting to observe that the space of Ω-trees maintains several statistical invariants although it is drastically reduced in size compared to the space of unconstrained Yule trees. Using generating functions, we study here some basic combinatorial properties of Ω-trees. We focus on the distribution of the number of subtrees with two leaves. We show that expectation and variance of this distribution match those for unconstrained trees already for very small values of ω.

Greedy Trees, Subtrees and Antichains

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree.

Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of $\sigma_T(w)/\sigma_T(u)$, $\sigma_T(w)/\sigma_T(v)$, $\sigma(T)/\sigma_T(v)$, and $\sigma(T)/\sigma_T(w)$, where $T$ is a tree on $n$ vertices, $v$ is in the centroid of the tree $T$, and $u,w$ are leaves in $T$.In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of $F_T(w)/F_T(u)$, $F_T(w)/F_T(v)$, $F(T)/F_T(v)$, and $F(T)/F_T(w)$, where $T$ is a tree on $n$ vertices, $v$ is in the subtree core of the tree $T$, and $u,w$ are leaves in $T$-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs.

RETRACTED ARTICLE: Strong limiting behavior in binary search trees

In a binary search tree of size n, each node has no more than two children, we denote the number of the node with k children by . In this paper, we study the strong limit behavior of the random variables and , where represents the number of subtrees of size m. The results can imply some known results. MSC: 60F05, 05C80.

The coalescent point process of branching trees

We define a doubly infinite, monotone labeling of Bienaymé–Galton–Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_{i};i\ge1)$, where $A_{i}$ is the coalescence time between individuals $i$ and $i+1$. There is a Markov process of point measures $(B_{i};i\ge1)$ keeping track of more ancestral relationships, such that $A_{i}$ is also the first point mass of $B_{i}$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation $h$ in a planar BGW tree conditioned to survive $h$ generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\mathbb{R}_{+}$ associated with the limiting continuous-state branching (CSB) process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting CSB population by considering only points with coalescence times greater than $\varepsilon $. The limiting coalescent point process $(B^{\varepsilon}_{i};i\ge1)$ is the sequence of depths greater than $\varepsilon$ of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. We prove that in the general case the coalescent process with multiplicities $(B^{\varepsilon}_{i};i\ge1)$ is a Markov chain of point masses and we give an explicit formula for its transition function. The paper ends with two applications in the discrete case. Our results show that the sequence of $A_{i}$’s are i.i.d. when the offspring distribution is linear fractional. Also, the law of Yaglom’s quasi-stationary population size for subcritical BGW processes is disintegrated with respect to the time to most recent common ancestor of the whole population.

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.

The vertex leafage of chordal graphs

Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model of G. This representation is not necessarily unique. The leafage ℓ(G) of a chordal graph G is the minimum number of leaves of the host tree of a tree model of G. The leafage is known to be polynomially computable.In this contribution, we introduce and study the vertex leafage. The vertex leafage vℓ(G) of a chordal graph G is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves. In particular, the case vℓ(G)≤2 coincides with the class of path graphs (vertex intersection graphs of paths in trees).We prove for every fixed k≥3 that deciding whether the vertex leafage of a given chordal graph is at most k is NP-complete. In particular, we show that the problem is NP-complete on split graphs with vertex leafage of at most k+1. We further prove that it is NP-hard to find for a given split graph G (with vertex leafage at most three) a tree model with minimum total number leaves in all subtrees, or where maximum number of subtrees are paths. On the positive side, for chordal graphs of leafage at most ℓ, we show that the vertex leafage can be calculated in time nO(ℓ).Finally, we prove that every chordal graph G admits a tree model that realizes both the leafage and the vertex leafage of G. Notably, for every path graph G, there exists a path model with ℓ(G) leaves in the host tree and we describe an O(n3) time algorithm to compute such a path model.