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Okoth, Isaac Owino. "Combinatorics of oriented trees and tree-like structures." Thesis, Stellenbosch : Stellenbosch University, 2015. http://hdl.handle.net/10019.1/96860.
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Thesis (PhD)--Stellenbosch University, 2015.ENGLISH ABSTRACT : In this thesis, a number of combinatorial objects are enumerated. Du and Yin as well as Shin and Zeng (by a different approach) proved an elegant formula for the number of labelled trees with respect to a given in degree sequence, where each edge is oriented from a vertex of lower label towards a vertex of higher label. We refine their result to also take the number of sources (vertices of in degree 0) or sinks (vertices of out degree 0) into account. We find formulas for the mean and variance of the number of sinks or sources in these trees. We also obtain a differential equation and a functional equation satisfied by the generating function for these trees. Analogous results for labelled trees with two marked vertices, related to functional digraphs, are also established. We extend the work to count reachable vertices, sinks and leaf sinks in these trees. Among other results, we obtain a counting formula for the number of labelled trees on n vertices in which exactly k vertices are reachable from a given vertex v and also the average number of vertices that are reachable from a specified vertex in labelled trees of order n. In this dissertation, we also enumerate certain families of set partitions and related tree-like structures. We provide a proof for a formula that counts connected cycle-free families of k set partitions of {1, . . . , n} satisfying a certain coherence condition and then establish a bijection between these families and the set of labelled free k-ary cacti with a given vertex-degree distribution. We then show that the formula also counts coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We extend the work to count coloured oriented cacti and coloured cacti. Noncrossing trees and related tree-like structures are also considered in this thesis. Specifically, we establish formulas for locally oriented noncrossing trees with a given number of sources and sinks, and also with given indegree and outdegree sequences. The work is extended to obtain the average number of reachable vertices in these trees. We then generalise the concept of noncrossing trees to find formulas for the number of noncrossing Husimi graphs, cacti and oriented cacti. The study is further extended to find formulas for the number of bicoloured noncrossing Husimi graphs and the number of noncrossing connected cycle-free pairs of set partitions.
AFRIKAANSE OPSOMMING : In hierdie tesis word ’n aantal kombinatoriese objekte geenumereer. Du en Yin asook Shin en Zeng (deur middel van ’n ander benadering) het ’n elegante formule vir die aantal geëtiketteerde bome met betrekking tot ’n gegewe ingangsgraadry, waar elke lyn van die nodus met die kleiner etiket na die nodus met die groter etiket toe georiënteer word. Ons verfyn hul resultaat deur ook die aantal bronne (nodusse met ingangsgraad 0) en putte (nodusse met uitgangsgraad 0) in ag te neem. Ons vind formules vir die gemiddelde en variansie van die aantal putte of bronne in hierdie bome. Ons bepaal verder ’n differensiaalvergelyking en ’n funksionaalvergelyking wat deur die voortbringende funksie van hierdie bome bevredig word. Analoë resultate vir geëtiketteerde bome met twee gemerkte nodusse (wat verwant is aan funksionele digrafieke), is ook gevind. Ons gaan verder voort deur ook bereikbare nodusse, bronne en putte in hierdie bome at te tel. Onder andere verkry ons ’n formule vir die aantal geëtiketteerde bome met n nodusse waarin presies k nodusse vanaf ’n gegewe nodus v bereikbaar is asook die gemiddelde aantal nodusse wat bereikbaar is vanaf ’n gegewe nodus. Ons enumereer in hierdie tesis verder sekere families van versamelingsverdelings en soortgelyke boom-vormige strukture. Ons gee ’n bewys vir ’n formule wat die aantal van samehangende siklus-vrye families van k versamelingsverdelings op {1, . . . , n} wat ’n sekere koherensie-vereiste bevredig, en ons beskryf ’n bijeksie tussen hierdie familie en die versameling van geëtiketteerde vrye k-êre kaktusse met ’n gegewe nodus-graad-verdeling. Ons toon ook dat hierdie formule ook gekleurde Husimi-grafieke tel waar blokke van dieselfde kleur nie insident met mekaar mag wees nie. Ons tel verder ook gekleurde georiënteerde kaktusse en gekleurde kaktusse. Nie-kruisende bome en soortgelyke boom-vormige strukture word in hierdie tesis ook beskou. On bepaal spesifiek formules vir lokaal georiënteerde nie-kruisende bome wat ’n gegewe aantal bronne en putte het asook nie-kruisende bome met gegewe ingangs- en uitgangsgraadrye. Ons gaan voort deur die gemiddelde aantal bereikbare nodusse in hierdie bome te bepaal. Ons veralgemeen dan die konsep van nie-kruisende bome en vind formules vir die aantal nie-kruisende Husimi-grafieke, kaktusse en georiënteerde kaktusse. Laastens vind ons ’n formule vir die aantaal tweegekleurde nie-kruisende Husimi-grafieke en die aantal nie-kruisende samehangende siklus-vrye pare van versamelingsverdelings.
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Drinkwater, Kara M. "Trees." VCU Scholars Compass, 2006. http://hdl.handle.net/10156/1907.
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Garton, Ian Spencer. "Concurrency in B-trees and tries : search and insert." Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=33400.
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Multiuser database systems require concurrency control in order to perform correctly. B-trees have become the standard data structure for storing indices that aid in data retrieval and there have been many algorithms published to enable concurrent operations for B-trees. Tries are another data structure useful for storing index data, particularly for text and spatial databases. Significant data compression can be achieved by using a trie to store index values. However, there have been no algorithms published to support concurrent trie operations.We present algorithms that enable concurrent searches and inserts for tries with pointerless representation. We also measure the performance of our algorithms and compare with that of the best B-tree algorithms. In order to measure trie concurrency, we survey a number of studies that have been made for B-tree concurrency. Using these published studies, we build a simulation model to measure the concurrency of our algorithms.
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Hall, Justine Michelle. "Trees in towns : factors affecting the distribution of trees in high density residential areas of Greater Manchester." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/trees-in-towns-factors-affecting-the-distribution-of-trees-in-high-density-residential-areas-of-greater-manchester(568b58f3-4524-4a8d-abba-2094c4e21567).html.
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The distribution of trees across urban areas of the UK has been shown to be uneven, with lower density residential areas containing many more trees and much higher tree cover than areas of higher density housing. However, in Greater Manchester, tree number within high density housing areas also varies substantially. This thesis sought to explore the reasons for this variation in tree cover, whether tree cover should be increased and if so, how. The research investigated a potential cause for the variation in number of trees and tree cover within high density housing areas – housing type – for the study area of western Greater Manchester. Eleven different types of high density housing were categorised and all high density housing within the study area was classified as one of these types. Within these housing types, the amount of tree cover was determined, along with the proportions of other surface types. The land uses where the trees were growing were also determined. Finally, the potential increases in tree cover were also calculated for each housing type by a simulated planting technique. Maximum surface temperatures and rainfall runoff were calculated using computer models, for both existing and potential tree cover in each housing type. It was found that urban tree cover varies from 1.6% in pre 1919 terraced housing that opens directly onto the road to 14.8% in 1960s walkway-style housing. Tree cover could theoretically be increased by at least 5% in all housing types, reducing maximum surface temperatures by at least 1°C. In housing types with less than 4% existing tree cover, maximum surface temperatures could be reduced by up to 4.5°C. The views of residents were determined using a postal questionnaire about urban trees sent to residents of 4 different types of street environment. Residents of all street types surveyed were very positive about urban trees; their attitudes were not affected by whether there are trees in their street or not. The vast majority of respondents considered trees important to their quality of life, and that cost to the council should not prevent tree planting. The views and practices around urban trees and greening by practitioners were determined by running a workshop and their recommendations to increase tree cover are presented. These include changes in funding to include money for tree maintenance after planting, the importance of a full tree inventory and innovative ways to raise funding for trees. The effectiveness of a community greening scheme at increasing tree cover was compared with two regeneration schemes. The community tree planting scheme was found to deliver tree planting much closer to the potential than regeneration schemes.
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Abu-Ata, Muad Mustafa. "Tree-Like Structure in Graphs and Embedability to Trees." Kent State University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1397345185.
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Bryant, David. "Building trees, hunting for trees, and comparing trees : theory and methods in phylogenetic analysis." Thesis, University of Canterbury. Mathematics and Statistics, 1997. http://hdl.handle.net/10092/7914.
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Phylogenetics is the study and identification of evolutionary patterns and structures in nature; this thesis explores the mathematics of these structures. The basic objects of study are the leaf labelled tree and its substructures: quartets, splits, clusters and rooted triples, among others. We present fundamental theorems and characterisations, as well as efficient algorithms for a range of phylogenetic problems.
It is often possible to deduce phylogenetic information not in the original data. We characterise an intriguing system of inference 'rules' that arise in this way, and prove that there exist rules of every order that cannot be reduced to lower order rules.
We describe a polynomial time algorithm that extracts maximum weight bounded degree trees from a given binary character set. The algorithm enables compatibility analysis of large data sets, in this case the daunting "Out of Africa" human mtDNA sequences. Other applications include consensus, quartet puzzling and split decomposition.
We accelerate the Minimum Evolution method with an optimal O(n²) time algorithm
for calculating OLS edge lengths and fast algorithms for WLS and GLS edge lengths. We show how a Minimum Evolution tree can be efficiently extracted from a collection of splits.
Consensus methods are surveyed, characterised and classified. A new intuitive consensus method for edge weighted trees is introduced, together with an efficient algorithm for constructing it.
We present an algorithm for the Maximum Agreement Subtree problem that is based on rooted triples and is much simpler than existing algorithms. We also provide algorithms for obtaining agreement subtrees with the largest number of edges, rooted triples or quartets.
Issues of complexity are discussed throughout the thesis, with several new NP-completeness results and a list of standard NP-complete phylogenetic problems.
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Balakrishnan, Anantaram, Thomas L. Magnanti, and Prakash Mirchandani. "Heuristics, LPs, and Generalizations of Trees on Trees." Massachusetts Institute of Technology, Operations Research Center, 1993. http://hdl.handle.net/1721.1/5377.
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We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a linear programming relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and linear programming relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, the performance guarantees for both the heuristic and linear programming relaxation of the combined overlay optimization model are 33%. We also develop heuristic and linear programming performance guarantees for specialized models, including a dual path connectivity model with a worst-case performance guarantee of 25%, and an uncapacitated multicommodity network design model with a worst-case performance guarantee (approximately) proportional to the square root of the number of commodities.
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Magnanti, Thomas L., and Laurence A. Wolsey. "Optimal Trees." Massachusetts Institute of Technology, Operations Research Center, 1994. http://hdl.handle.net/1721.1/5122.
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Prosser, Christopher Skinner 1978. "Two Trees." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/11048.
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1 score (viii, 79 p.) A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.The Two Trees is a fifteen-minute musical composition for orchestra. Inspired by William Butler Yeats' poem of the same name, the piece depicts the images described by Yeats' poetic narrative through a double theme and variations form consisting of two contrasting themes that are related, one ascending and one descending. Each theme represents one of the two contrasting sections of the poem and is followed by a set of five variations for a total of ten. Since the rhyme scheme of each section of the poem is divided into five phrases of four lines, each musical variation corresponds to four lines of text.
Committee in Charge: David Crumb, Chair; Robert Kyr; Jack Boss
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Taylor, Paul Clifford. "Classification trees." Thesis, University of Bath, 1990. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.306312.
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Balakrishnan, Anantaram, Thomas L. Magnanti, and Prakash Mirchandani. "Heuristics, LPs, and Trees on Trees: Network Design Analyses." Massachusetts Institute of Technology, Operations Research Center, 1994. http://hdl.handle.net/1721.1/5277.
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We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a LP relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and LP relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, both the heuristic and the LP relaxation of the combined overlay optimization model have performance guarantees of 4/3. We extend this analysis to multiple overlays on the same base solution, producing the first known worst-case bounds (approximately proportional to the square root of the number of commodities) for the uncapacitated multicommodity network design problem. In a companion paper, we develop heuristic performance guarantees for various new multi-tier. survivable network design models that incorporate both multiple facility types or technologies and differential node connectivity levels.
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Choi, Myung Jin Ph D. Massachusetts Institute of Technology. "Trees and beyond : exploiting and improving tree-structured graphical models." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/62888.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student submitted PDF version of thesis.
Includes bibliographical references (p. 169-179).
Probabilistic models commonly assume that variables are independent of each other conditioned on a subset of other variables. Graphical models provide a powerful framework for encoding such conditional independence structure of a large collection of random variables. A special class of graphical models with significant theoretical and practical importance is the class of tree-structured graphical models. Tree models have several advantages: they can be easily learned given data, their structures are often intuitive, and inference in tree models is highly efficient. However, tree models make strong conditional independence assumptions, which limit their modeling power substantially. This thesis exploits the advantages of tree-structured graphical models and considers modifications to overcome their limitations. To improve the modeling accuracy of tree models, we consider latent trees in which variables at some nodes represent the original (observed) variables of interest while others represent the latent variables added during the learning procedure. The appeal of such models is clear: the additional latent variables significantly increase the modeling power, and inference on trees is scalable with or without latent variables. We propose two computationally efficient and statistically consistent algorithms for learning latent trees, and compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree models. We exploit the advantages of tree models in the application of modeling contextual information of an image. Object co-occurrences and spatial relationships can be important cues in recognizing and localizing object instances. We develop tree-based context models and demonstrate that its simplicity enables us to integrate many sources of contextual information efficiently. In addition to object recognition, we are interested in using context models to detect objects that are out of their normal context. This task requires precise and careful modeling of object relationships, so we use a latent tree for object co-occurrences. Many of the latent variables can be interpreted as scene categories, capturing higher-order dependencies among object categories. Tree-structured graphical models have been widely used in multi-resolution (MR) modeling. In the last part of the thesis, we move beyond trees, and propose a new modeling framework that allows additional dependency structure at each scale of an MR tree model. We mainly focus on MR models with jointly Gaussian variables, and assume that variables at each scale have sparse covariance structure (as opposed to fully-uncorrelated structure in MR trees) conditioned on variables at other scales. We develop efficient inference algorithms that are partly based on inference on the embedded MR tree and partly based on local filtering at each scale. In addition, we present methods for learning such models given data at the finest scale by formulating a convex optimization problem.
by Myung Jin Choi.
Ph.D.
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Skorin-Kapov, Darko. "On some problems on k-trees and partial k-trees." Thesis, University of British Columbia, 1989. http://hdl.handle.net/2429/29288.
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The objective of this thesis is to investigate some structural and algorithmic properties of k-trees and partial k-trees. A k-tree can be constructed from a k-complete graph by recursively adding a new vertex which is adjacent to all vertices of an existing k-complete subgraph. Partial k-trees are graphs embeddable in a k-tree with the same vertex set. They are natural generalizations of forests and series-parallel graphs which are the first two members of the hierarchy of partial k-trees. The many applications of k-trees and partial k-trees have motivated their study from both an algorithmic and a theoretical point of view. For example, k-trees arise in reliable communication network design problems (Farley (1981), Farley and Proskurowski (1982), Neufeld and Colbourn (1983), Wald and Colbourn (1983), Colbourn and Proskurowski (1984)) and in the study of the complexity of certain type of queries in a relational data base system (Arnborg (1979)). Moreover, the class of k-trees is special in the sense that many problems, which are NP-complete for arbitrary graphs, are solvable in polynomial time when restricted to k-trees or partial k-trees (Arnborg and Proskurowski (1989)).
In Chapter 2 of the thesis we analyze a fixed cost spanning forest (FCSF) problem, defined over a graph G, in which some customers require service that can be generated at some facilities' sites. Both the set of customers and facilities' sites are represented by nodes in G. There is a fixed cost for opening each facility and a cost for delivering the service from open facilities to the customers. Customers do not necessarily have to receive the service directly from an open facility, but possibly through other intermediate customers. We develop a linear time algorithm for solving the FCSF problem when the customers and potential facilities' sites are located on a series-parallel network or, equivalently, a partial 2-tree. We further analyze a related cost allocation problem, in which we seek a fair method for allocating the cost of providing the service to the customers. We formulate this cost allocation problem as a cooperative game and show that, in general, the core of this cooperative game may be empty. However, we provide a sufficient condition, which can be verified in polynomial time, for the nonemptiness of the core of this game.
A k-tree can be reduced to the k-complete graph by sequentially removing k-degree vertices with completely connected neighbors. We use this reduction process to develop, in Chapter 3, efficient algorithms for several optimization problems on k-trees and partial k-trees. In particular, we develop a linear time algorithm to find shortest simple paths from a given vertex to all other vertices in a k-tree, we compute the diameter of a k-tree with equal edge lengths in linear time, and we construct an O(n[sup k+2]) algorithm to solve the Simple Plant Location problem in an n-vertex partial k-tree.
In Chapter 4 we present a new characterization of a k-path between two vertices u and v, in an equal weight k-tree G, by means of minimal k and k+1 cliques with respect to certain partial orders defined on the collections of all k and k+1 cliques in a k-tree. We use it to develop an O(n²) algorithm to decompose a vertex set V of a k-tree G to a minimum number of components, such that for any pair of vertices i and j in the same component, the cable distance between i and j is bounded by a positive integer R. We also compute the k-cable diameter of a k-tree with equal edge lengths in linear time.
In Chapter 5 we derive some separation properties of partial k-trees and use them to develop NC algorithms for recognizing partial 2-trees and 3-trees. Explicitly, we prove the existence of a k-separator in a partial k-tree graph and construct a linear time algorithm that finds such a separator in k-trees. This algorithm can be used to obtain a balanced binary decomposition of a k-tree in 0(n log n) time. We derive some other separation properties of partial k-trees and use them to construct a balanced decomposition of an embedding of a k-connected partial k-tree when k = 2 and 3. Finally, we construct NC algorithms for the recognition of a partial k-tree for k - 2 and 3, which run in O(log²n) time using, respectively, O(n³) and O(n⁴) processors.Business, Sauder School of
Graduate
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Travis, David J., Vernon Meentemeyer, and Roger P. Belanger. "Stressed Trees Produce a Better Climatic Signal than Healthy Trees." Tree-Ring Society, 1990. http://hdl.handle.net/10150/261930.
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The basis for the selection of trees to be used in the production of dendrochronologies has long been an issue (Douglass 1946; Fritts 1976). In humid regions the common practice has been to use trees that appear to be in good health. As a part of a larger study involving the impact of ice storms on tree-ring increments (Travis 1989), we show that trees stressed as a result of ice damage produced a stronger climatic signal than nondamaged trees.
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Kirkby, Richard Brendon. "Improving Hoeffding Trees." The University of Waikato, 2008. http://hdl.handle.net/10289/2568.
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Modern information technology allows information to be collected at a far greater rate than ever before. So fast, in fact, that the main problem is making sense of it all. Machine learning offers promise of a solution, but the field mainly focusses on achieving high accuracy when data supply is limited. While this has created sophisticated classification algorithms, many do not cope with increasing data set sizes. When the data set sizes get to a point where they could be considered to represent a continuous supply, or data stream, then incremental classification algorithms are required. In this setting, the effectiveness of an algorithm cannot simply be assessed by accuracy alone. Consideration needs to be given to the memory available to the algorithm and the speed at which data is processed in terms of both the time taken to predict the class of a new data sample and the time taken to include this sample in an incrementally updated classification model. The Hoeffding tree algorithm is a state-of-the-art method for inducing decision trees from data streams. The aim of this thesis is to improve this algorithm. To measure improvement, a comprehensive framework for evaluating the performance of data stream algorithms is developed. Within the framework memory size is fixed in order to simulate realistic application scenarios. In order to simulate continuous operation, classes of synthetic data are generated providing an evaluation on a large scale. Improvements to many aspects of the Hoeffding tree algorithm are demonstrated. First, a number of methods for handling continuous numeric features are compared. Second, tree prediction strategy is investigated to evaluate the utility of various methods. Finally, the possibility of improving accuracy using ensemble methods is explored. The experimental results provide meaningful comparisons of accuracy and processing speeds between different modifications of the Hoeffding tree algorithm under various memory limits. The study on numeric attributes demonstrates that sacrificing accuracy for space at the local level often results in improved global accuracy. The prediction strategy shown to perform best adaptively chooses between standard majority class and Naive Bayes prediction in the leaves. The ensemble method investigation shows that combining trees can be worthwhile, but only when sufficient memory is available, and improvement is less likely than in traditional machine learning. In particular, issues are encountered when applying the popular boosting method to streams.
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Clark, David. "Compact PAT trees." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq21335.pdf.
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Kustra, Rafal. "Soft decision trees." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq28745.pdf.
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Kutscheid, Gloria J. "More than trees." Virtual Press, 1986. http://liblink.bsu.edu/uhtbin/catkey/473196.
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The purpose of this study was to complete a series of photographs utilizing trees as metaphors. This series of photographs is composed into book form for possible publication. It presents the psychological and symbolic impact of trees in different environments and from different perspectives through the development of a unique style of creative composition. The format consists of text and photographs. This project develops a working definition of metaphors and an approach using trees as the medium to convey various symbolic and psychological references. Both associated and inferred types of symbolisms are used. This type of approach is not necessarily unique and has been evidenced in the work of such "populist" poets as Robert Frost and the late photographer-poet Gordon Parks. With the basic inventory of symbols and psychological reference associations, whole trees, parts of trees, and/or groups of trees are identified to illustrate through the photographic medium that metaphors do indeed exist and can be found in trees. Photographs are produced, edited and composed to produce not only pleasing aesthetics, but also to provide continuity throughout the series. Photographs vary in size, shape, contrast, and composition to provide variation in the format.
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Puch-Solis, Roberto O. "Hierarchical junction trees." Thesis, University of Warwick, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365243.
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Della-Libera, Giovanni M. (Giovanni Moises). "Dynamic diffracting trees." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/40614.
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Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1997.Includes bibliographical references (p. 109-111).
by Giovanni M. Della-Libera.
M.Eng.
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Woznica, Hilary. "Hearts and Trees." VCU Scholars Compass, 2012. http://scholarscompass.vcu.edu/etd/306.
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The process of creating art is a visceral experience for me. Each piece is born from an inherent need to visually express and understand my experiences and emotions. I use a variety of materials to create mixed media sculptures, books, and two-dimensional images. Maps and trees are the common motifs and metaphors found in my work. I paint on the surface of maps to reference significant locations in my journey of personal growth. I incorporate found sticks and branches in my sculptures and paint images of trees to communicate my reverence for nature. I often use the tree as a symbol for myself. I create images and detail with careful control of my media. I use pliers to manipulate wire into precisely formed curls around stones and wood. I apply short, repeated strokes with pen or paint on paper, or with a ceramic tool on a clay surface. I use this repetition of marks to create visual rhythm and a sense of movement which gives energy to otherwise simple images. I use a limited color palette to help maintain visual clarity of the image.
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Wright, Glenn C. "Irrigating Citrus Trees." College of Agriculture and Life Sciences, University of Arizona (Tucson, AZ), 2000. http://hdl.handle.net/10150/144778.
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5 pp.Water is essential for citrus trees because it is an integral component of the biochemical reactions that occur within the plant, and also because it is the carrier that moves plant nutrients and other substances throughout the tree. This publication examines irrigating citrus trees. Topics include; water stress, estimating tree water use, timing water applications, dealing with salty water, calculating how much water is applied, as well as application methods and mulches. And it also includes a table which can be applied to spreadsheets, to calculate the exact amount of water a citrus tree needs.
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Wright, Glenn C. "Budding Citrus Trees." College of Agriculture and Life Sciences, University of Arizona (Tucson, AZ), 2000. http://hdl.handle.net/10150/144814.
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3 pp.Citrus budding is a plant propagation technique that any homeowner can do. Once the technique is learned, homeowners can add citrus tree. This publication addresses the budding techniques of citrus trees. Topics include the preparation prior to budding, selecting budsticks, storing budsticks, selecting and preparing the budding location, cutting the bud, inserting the bud in the t cut, and forcing the bud.
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Dawson, Shelly Jean. "Minimal congestion trees." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3005.
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Analyzes the results of M.I. Ostrovskii's theorem of inequalities which estimate the minimal edge congestion for finite simple graphs. Uses the generic results of the theorem to examine and further reduce the parameters of inequalities for specific families of graphs, particularly complete graphs and complete bipartite graphs. Also, explores a possible minimal congestion tree for some grids while forming a conjecture for all grids.
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Goudjil, Amar. "Data structures, binary search trees, a study of random Weyl trees." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0026/MQ50778.pdf.
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Goudjil, Amar. "Data structures, binary search trees : a study of random Weyl trees." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=21559.
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This thesis covers the study of a particular class of binary search trees, the Weyl trees formed by consecutive insertion of numbers {theta}, {2theta}, {3theta}, ..., where theta is an irrational number from (0,1), and {x} denotes the fractional part of x. Various properties of the structure of these trees are explored and a relationship with the continued fraction expansion of theta is shown. Among these properties, an approximation of the height Hn of a Weyl tree with n nodes is given when theta is chosen at random and uniformly on (0, 1). The main result of this work is that in probability, Hn ∼ (12/pi2) log n log log n.
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Holmgren, Cecilia. "Split Trees, Cuttings and Explosions." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-112239.
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This thesis is based on four papers investigating properties of split trees and also introducing new methods for studying such trees. Split trees comprise a large class of random trees of logarithmic height and include e.g., binary search trees, m-ary search trees, quadtrees, median of (2k+1)-trees, simplex trees, tries and digital search trees. Split trees are constructed recursively, using “split vectors”, to distribute n “balls” to the vertices/nodes. The vertices of a split tree may contain different numbers of balls; in computer science applications these balls often represent “key numbers”. In the first paper, it was tested whether a recently described method for determining the asymptotic distribution of the number of records (or cuts) in a deterministic complete binary tree could be extended to binary search trees. This method used a classical triangular array theorem to study the convergence of sums of triangular arrays to infinitely divisible distributions. It was shown that with modifications, the same approach could be used to determine the asymptotic distribution of the number of records (or cuts) in binary search trees, i.e., in a well-characterized type of random split trees. In the second paper, renewal theory was introduced as a novel approach for studying split trees. It was shown that this theory is highly useful for investigating these types of trees. It was shown that the expected number of vertices (a random number) divided by the number of balls, n, converges to a constant as n tends to infinity. Furthermore, it was demonstrated that the number of vertices is concentrated around its mean value. New results were also presented regarding depths of balls and vertices in split trees. In the third paper, it was tested whether the methods of proof to determine the asymptotic distribution of the number of records (or cuts) used in the binary search tree, could be extended to split trees in general. Using renewal theory it was demonstrated for the overall class of random split trees that the normalized number of records (or cuts) has asymptotically a weakly 1-stable distribution. In the fourth paper, branching Markov chains were introduced to investigate split trees with immigration, i.e., CTM protocols and their generalizations. It was shown that there is a natural relationship between the Markov chain and a multi-type (Galton-Watson) process that is well adapted to study stability in the corresponding tree. A stability condition was presented to describe a phase transition deciding when the process is stable or unstable (i.e., the tree explodes). Further, the use of renewal theory also proved to be useful for studying split trees with immigration. Using this method it was demonstrated that when the tree is stable (i.e., finite), there is the same type of expression for the number of vertices as for normal split trees.
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Kittle, T. Rene. "Influence of mycorrhizal inoculation treatments on native tree and shrub biomass and survival in a floodplain, Flathead Indian Reservation." CONNECT TO THIS TITLE ONLINE, 2007. http://etd.lib.umt.edu/theses/available/etd-06042007-104842/.
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Rathfon, Ronald A. "Developing DRIS norms for Fraser fir Christmas trees." Thesis, Virginia Tech, 1990. http://hdl.handle.net/10919/41610.
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Fraser fir [Abies fraseri (Pursh) Poir.] is an important Christmas tree species in Virginia. Because it is responsive to fertilization, and because most Fraser fir growers fertilize their crop, a scientifically-based nutrient diagnosis and fertilizer recommendation system is needed. The objective of this study was to develop and test DRIS norms for Fraser fir Christmas trees grown in Virginia for the ultimate purpose of establishing a nutrition diagnosis and fertilizer prescription system. A total of 107 Fraser fir plantations were sampled for foliage, soil, and diameter measurements. These plantations represented the range in site conditions and management practices for Fraser fir Christmas trees grown in Virginia. Foliage was analyzed for N, P, K, Ca, Mg, Zn, Mn, Fe, Cu, and B. Soil was analyzed for extractable macro- and micronutrients. DRIS norms were developed from these data using standard DRIS procedures. A total of 42 nutrient ratios were significant discriminators of tree performance as measured by variation in ground line diameter. The norms were tested using sixth year data from a factorial fertilizer trial. Nutrient limitations due to both deficiencies and imbalance were detected and correctly diagnosed using the newly-derived norms. A complete validation is required, but this preliminary test showed that these norms are useable and useful in their present form. In the process of developing and testing the norms, modifications to traditional DRIS methods were used to meet the special conditions of this crop. DRIS symmetry was maintained by including non-significant ratios, but setting their standardization functions equal to zero. This reduced the influence of the non-discriminating nutrient ratios on the DRIS analysis. Norm ranges as opposed to discrete norms (ratio means) were used to correct for the influence of extremely variable micronutrient ratios on the DRIS analysis. Soil norms did not enhance diagnoses over and above using foliar norms alone. This is due to soil sampling variation, poor correlations of extractable nutrients with tree performance, and an incomplete understanding of fertilizer reactions and uptake chemistry in a variety of soils. Each crop presents unique challenges in the application of DRIS. DRIS should not be naively applied without investigating these problems. The DRIS norms established in this study, and the modifications to standard DRIS methods, provide a sound scientific basis upon which to build a nutrient diagnosis and fertilizer recommendation system for Fraser fir Christmas trees grown in Virginia.Master of Science
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Riitters, Kurt H. "Analysis of Biweight Site Chronologies: Relative Weights of Individual Trees over Time." Tree-Ring Society, 1990. http://hdl.handle.net/10150/261911.
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The relative weights on individual trees in a biweight site chronology can indicate the consistency of tree growth responses to macroclimate and can be the basis for stratifying trees in climate-growth analyses. This was explored with 45 years of ring-width indices for 200 trees from five even-aged jack pine (Pints banksiana Lamb.) stands. Average individual-tree relative weights were similar, but most trees had at least one transient occurrence of low relative weight. The standard deviations of individual-tree relative weights suggested that some trees had mom variable growth responses than others. The trees were classified by the average and standard deviation of their relative weights, and biweight site chronologies were then calculated for these subgroups. Chronologies derived from trees with low average weights, and from trees with high standard deviation of weights, sometimes appeared to be different from chronologies derived from the remaining trees.
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Biondi, Franco. "Development of a Tree-Ring Network for the Italian Peninsula." Tree-Ring Society, 1992. http://hdl.handle.net/10150/262358.
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This article describes the analysis of tree-ring collections from standing trees of sixteen species at twenty sites distributed throughout the Italian Peninsula. Visual and numerical crossdating among ring widths allowed the computation of standard and residual tree-ring chronologies. Relationships among chronologies were identified by Spearman's coefficient of rank correlation, using Bonferroni's inequality to adjust significance level. The oldest living tree sampled to date is a 963-year old palebark pine (Pinus leucodermis Ant.) at Parco del Pollino. Individuals more than two centuries old were identified at eleven sites for eight species. The tree-ring network so far consists of twenty-two chronologies for nine species at nineteen sites. Seven conifer species account for ten chronologies and two angiosperm species account for the remaining twelve chronologies. The most represented species is Fagus sylvatica L., with eleven chronologies distributed over the entire peninsula and highly correlated with one another. The order of autoregressive models fitted to the data never exceeded two. In particular, the order of autoregressive models fitted to Fagus sylvatica chronologies decreased with decreasing age of sampled trees. Based on the significant coefficients of rank correlation, residual chronologies of Fagus sylvatica could be separated into northern, central, and southern groups. This points to the existence of broad regions distributed along a latitudinal gradient, corresponding to large-scale climatic regimes over the Italian Peninsula.
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Grissino-Mayer, Henri D. "An Updated List of Species Used in Tree-Ring Research." Tree-Ring Society, 1993. http://hdl.handle.net/10150/262378.
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During the past 100 years, researchers have investigated the potential of hundreds of tree and shrub species for use in applications of tree-ring research. Although several lists of species known to crossdate have been published, investigated species that do not crossdate are rarely included despite the usefulness of this information for future research. This paper provides a list of the Latin and common names of 573 species that have been investigated in tree-ring research, information on species known to crossdate, and information on species with measurement and/or chronology data in the International Tree-Ring Data Bank. In addition, a measure of the suitability of a species for future tree-ring applications, the Crossdating Index (CDI), is developed and proposed for standard usage.
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Thia, Joshua A. Y. W. "The plight of trees in disturbed forest: conservation of Montane Trees, Nigeria." Thesis, University of Canterbury. School of Biological Sciences, 2014. http://hdl.handle.net/10092/9887.
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The montane forests of Africa represent some of the Earth's most diverse and threatened ecosystems. In particular, those in West Africa have received comparatively little attention from scientists in terms of understanding the ecology and biodiversity of their species. This thesis wishes to understand genetic and ecological factors that underpin the long-term survival of selected tree species (Cordia millenii, Entandrophragma angolense, Lovoa trichilioides) in the montane forests of the Mambilla Plateau, Nigeria. The results obtained here provide a strong foundation for future work that wishes to preserve the diverse forests of this region.
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Tadros, Alexandre. "Topological recursive fitting trees : A framework for interpretable regression extending decision trees." Thesis, KTH, Skolan för elektroteknik och datavetenskap (EECS), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-272130.
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Many real-world machine learning applications need interpretation of an algorithm output. The simplicity of some of the most fundamental machine learning algorithms for regression, such as linear regression or decision trees, facilitates interpretation. However, they fall short when facing complex (e.g. high-dimensional, nonlinear) relationships between variables. Several approaches like artificial neural networks and bagging or boosting variants of decision trees have been able to overcome this issue but at the cost of interpretation. We propose a framework called Topological Recursive Fitting (TRF) in which a decision tree is learned based on topological properties of the data. We expect the tree structure of our approach to enable interpretation while achieving comparable performance to previously mentioned blackbox methods. Results show that TRF can achieve comparable performance with those methods, even if confirmation on a more significant number of datasets should be initiated.Många verkliga maskininlärningsapplikationer behöver tolkning av resultatet från en algoritm. Enkelheten i några av de mest grundläggande maskininlärningsalgoritmerna för regression, såsom linjär regression eller beslutsträd, underlättar tolkningen. Men det räcker inte till när de möter komplexa (t.ex. högdimensionella, icke-linjära) förhållanden mellan variabler. Flera tillvägagångssätt som konstgjorda neurala nätverk och bagging eller att öka varianter av beslutsträd har kunnat övervinna denna fråga, men på bekostnad av tolkningen. Vi föreslår ett ramverk som heter Topologiskt Rekursiva Anpassning (Topological Recursive Fitting, TRF) där ett beslutsträd lärs utifrån datats topologiska egenskaper. Vi förväntar oss att trädstrukturen i vårt tillvägagångssätt möjliggör tolkning och samtidigt uppnår jämförbar prestanda med tidigare nämnda black-box-metoder. Resultaten visar att TRF kan uppnå jämförbar prestanda med dessa metoder, även om bekräftelse på ett mer betydande antal datamängder bör initieras.
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Yao, Chu. "Quad general tree drawing algorithm and general trees characterization : towards an environment for the experimental study on general tree drawing algorithms /." Full text available online, 2008. http://www.lib.rowan.edu/find/theses.
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MacKinnon, Richard Kyle. "Seeing the forest for the trees: tree-based uncertain frequent pattern mining." Springer International Publishing, 2014. http://hdl.handle.net/1993/31059.
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Many frequent pattern mining algorithms operate on precise data, where each data point is an exact accounting of a phenomena (e.g., I have exactly two sisters). Alas, reasoning this way is a simplification for many real world observations. Measurements, predictions, environmental factors, human error, &ct. all introduce a degree of uncertainty into the mix. Tree-based frequent pattern mining algorithms such as FP-growth are particularly efficient due to their compact in-memory representations of the input database, but their uncertain extensions can require many more tree nodes. I propose new algorithms with tightened upper bounds to expected support, Tube-S and Tube-P, which mine frequent patterns from uncertain data. Extensive experimentation and analysis on datasets with different probability distributions are undertaken that show the tightness of my bounds in different situations.February 2016
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Horton, M. "Graceful Trees: Statistics and Algorithms." Thesis, Honours thesis, University of Tasmania, 2003. https://eprints.utas.edu.au/19/1/GracefulTreesStatisticsAndAlgorithms.pdf.
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The Graceful Tree Conjecture is a problem in graph theory that dates back to 1967. It suggests that every tree on n nodes can be labelled with the integers [1..n] such that the edges, when labelled with the difference between their endpoint node labels, are uniquely labelled with the integers [1..n-1]. To date, no proof or disproof of the conjecture has been found, but all trees with up to 28 vertices have been shown to be graceful. The conjecture also leads to a problem in algorithm design for efficiently finding graceful labellings for trees. In this thesis, a new graceful labelling algorithm is described and used to show that all trees on 29 vertices are graceful. A study is also made of statistical trends in the proportion of tree labellings that are graceful. These trends offer strong additional evidence that every tree is graceful.
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Kühn, Daniela. "Cycles, minors and trees." [S.l.] : [s.n.], 2001. http://www.sub.uni-hamburg.de/disse/511/kuehn.pdf.
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König, Bernhard. "Trees, games and reflections." [S.l.] : [s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=964459620.
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Strothmann, Willy-B. "Bounded degree spanning trees /." Paderborn : HNI, 1997. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=007959737&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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König, Bernhard Andreas. "Trees, Games and Reflections." Diss., lmu, 2002. http://nbn-resolving.de/urn:nbn:de:bvb:19-81.
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Brumpton, Aaron M. W. "Molecules, trees, and guillotines." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ28178.pdf.
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Zamora, Cura Carlos Alberto. "Analysis of random trees." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape3/PQDD_0032/NQ64705.pdf.
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Cole, July Oskar. "Red Trees and Why." The University of Montana, 2010. http://etd.lib.umt.edu/theses/available/etd-06032010-203108/.
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Zhao, Rongyao. "Placements of euclidean trees." Thesis, McGill University, 1989. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74356.
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We define a Euclidean tree T = $(V,E)$ to be a weighted tree such that each edge $e in E$ has been assigned a positive integer weight l(e) called the length of e. A placement of T = $(V,E)$ is an assignment $f: V to { bf R} sp2$ (or R$ sp3$) of vertices to points in space satisfying the following: for each edge $e = (u,v) in E$, the Euclidean distance between f(u) and f(v) is equal to l(e). We define the EUCLIDEAN TREE PLACEMENT problem as follows. Given a Euclidean tree T = (V,E) and a subset of m vertices in V labeled $u sb1,u sb2, cdots,u sb{m}$, and given m nonempty sets $S sb1,S sb2, cdots,S sb{m}$, not necessarily disjoint, in R$ sp2$ (or R$ sp3$), find the feasible placements $f: V to { bf R} sp2$ (or R$ sp3$) such that for $1 le i le m$, $f(u sb{i}) in S sb{i}$, or else report no such placement exists.In this thesis, we study the EUCLIDEAN TREE PLACEMENT problem and give both lower and upper bounds for it. Specifically, we first examine two special decision versions of the problem and show their NP-hardness by reductions from PARTITION. Then we provide a geometric method for placing Euclidean trees in the plane with each of the leaves constrained to a point. We also discuss an algebraic method for doing this. The given constraints on the leaves implicitly impose constraints on all vertices in V. Thus each vertex is associated with a point set (not necessarily connected) in which it can be placed in some feasible placement of the tree. Each such point set is shown to have a finite number of circular arcs on its boundary. Let r be the maximum boundary arc number over all such point sets for a Euclidean tree having at most n vertices. Then, for an input tree with n vertices, the geometric method takes $O(nr sp2 log r)$ time (real RAM), and the algebraic method takes $O(2 sp{4n sp2})$ time (rational RAM). At present r is not known to be bounded by a polynomial in n. Nevertheless, the geometric algorithm seems much more practical to implement than the algebraic algorithm.
In developing the geometric method, we design an $O(m log sp2m)$ algorithm for computing the Minkowski sum of a planar figure, possibly containing holes and bounded by a collection of m circular arcs of various radii, with a circle centered at the origin. This algorithm can be directly used for computing the offset of the figure.
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Collins, Karen L. (Karen Linda). "Distance matrices of trees." Thesis, Massachusetts Institute of Technology, 1986. http://hdl.handle.net/1721.1/77687.
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Pardi, Fabio. "Algorithms on phylogenetic trees." Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.611685.
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Razanajatovo, Misanantenaina Valisoa. "Properties of greedy trees." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/95909.
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Thesis (MSc)--Stellenbosch University, 2014.ENGLISH ABSTRACT: A greedy tree is constructed from a given degree sequence using a simple greedy algorithm that assigns the highest degree to the root, the second, the third, . . . , -highest degree to the root’s neighbours, etc. This particular tree is the solution to numerous extremal problems among all trees with given degree sequence. In this thesis, we collect results for some distancebased graph invariants, the number of subtrees and the spectral radius in which greedy trees play a major role. We show that greedy trees are extremal for the aforementioned graph invariants by means of two different approaches, one using level greedy trees and majorization, while the other one is somewhat more direct. Finally, we prove some new results on greedy trees for additive parameters with specific toll functions.
AFRIKAANSE OPSOMMING: ’n Gulsige boom word vanuit ’n gegewe graadry deur middel van ’n eenvoudige gulsige algoritme gebou, wat die hoogste graad aan die wortel toewys, die tweede-, derde-, . . . , -hoogste graad aan die wortel se bure, ens. Hierdie spesifieke boom is die oplossing van ’n groot aantal ekstremale probleme onder al die bome met gegewe graadry. In hierdie tesis beskou ons ’n versameling van resultate oor afstand-gebaseerde grafiekinvariante, die aantal subbome en die spektraalstraal waarin gulsige bome ’n belangrike rol speel. Ons bewys dat gulsige bome ekstremaal vir die bogenoemde grafiekinvariante is deur van twee verskillende tegnieke gebruik te maak: een met behulp van vlak-gulsige bome en majorering, en ’n ander metode wat effens meer direk is. Laastens bewys ons sommige nuwe resultate oor gulsige bome vir additiewe parameters met spesifieke tolfunksies.
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Riveros, Jaeger Cristian. "Repairing strings and trees." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:012d384f-d1d0-471b-ae6e-bbf337892680.
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What do you do if a computational object fails a specification? An obvious approach is to repair it, namely, to modify the object minimally to get something that satisfies the constraints. In this thesis we study foundational problems of repairing regular specifications over strings and trees. Given two regular specifications R and T we aim to understand how difficult it is to transform an object satisfying R into an object satisfying T. The setting is motivated by considering R to be a restriction -- a constraint that the input object is guaranteed to satisfy -- while T is a target -- a constraint that we want to enforce. We first study which pairs of restriction and target specifications can be repaired with a ``small'' numbers of changes. We formalize this as the bounded repair problem -- to determine whether one can repair each object satisfying R into T with a uniform number of edits. We provide effective characterizations of the bounded repair problem for regular specifications over strings and trees. These characterizations are based on a good understanding of the cyclic behaviour of finite automata. By exploiting these characterizations, we give optimal algorithms to decide whether two specifications are bounded repairable or not. We also consider the impact of limitations on the editing process -- what happens when we require the repair to be done sequentially over serialized objects. We study the bounded repair problem over strings and trees restricted to this streaming setting and show that this variant can be characterized in terms of finite games. Furthermore, we use this characterization to decide whether one can repair a pair of specifications in a streaming fashion with bounded cost and how to obtain a streaming repair strategy in this case. The previous notion asks for a uniform bound on the number of edits, but having this property is a strong requirement. To overcome this limitation, we study how to calculate the maximum number of edits per character needed to repair any object in R into T. We formalize this as the asymptotic cost -- the limit of the number of edits divided by the length of the input in the worst case. Our contribution is an algorithm to compute the asymptotic cost for any pair of regular specifications over strings. We also consider the streaming variant of this cost and we show how to compute it by reducing this problem to mean-payoff games.
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Kemerling, Robert Alan. "Controlling with Model Trees." NSUWorks, 2011. http://nsuworks.nova.edu/gscis_etd/193.
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This dissertation develops a method of control for nonlinear processes based on regression trees with kernel regression at the leaves as a general control methodology. This methodology offers the ability to control a wide variety of processes exhibiting nonlinear behavior. It takes a place with other machine learning methods that are being applied to nonlinear control, but it does not suffer from the shortcomings of other methods. The method draws on two well-known machine learning methods, regression trees and kernel regression. This dissertation shows that this control method may be programmed from supervised runs of a process and it may be updated from follow-on runs. The method may serve as a backup controller for redundancy or it may be the primary controller for a process after supervised training by an expert operator.
Model trees are a generalized form of regression tree where the leaves contain more than one data point and a small kernel regression is performed when a leaf is selected by navigation through the tree. Kernel regression is a form of instance-based learning where the prediction is formed by distance-weighted input of the data. Kernel regression has been shown to fit complex, nonlinear forms quite well, but it suffers from scalability to higher dimensions as a potential control method. Using kernel regression with the innovative approach of constructing the model tree in a way to create a predicted control setting is the contribution of this research to the area of nonlinear control.
During this dissertation, the performance of model trees for situations of nonlinear control was demonstrated in four widely-varying settings. In all cases, the model tree was developed from process data in a common form. Where data were available to both develop the model tree and compare its prediction to retained test data, the model tree was able to demonstrate control prediction. Statistical tests of hypotheses failed to reject a null hypothesis of equal means and variances. Where the model tree could be created from a simulation and then used as a predictive control, the results demonstrated that the model tree could be used as a controller to meet process goals.
The demonstrations in this research displayed many different forms of nonlinearity that may be encountered in processes. In addition to typical nonlinear factors such as power equations and trigonometric functions, these demonstrations exposed the model tree to sudden, non-deterministic inputs and changes to inputs. Inputs that represent integer multiples of power sources are experienced. In some processes, the system includes some regions of linear response, but the boundaries of a region have a sudden change of slope or even an inflection point. Model trees with kernel regression at the leaves demonstrated the ability to handle these scenarios.