DOCSLIB.ORG

Tree a Tree Example More Example

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tree a Tree Example More Example A Tree • Definition – A is a connected undirected graph with no Tree simple circuit Section 10.1 prepared by Nattee Niparnan Example More Example • Family Tree prepared by Nattee Niparnan prepared by Nattee Niparnan A Forest Theorem 1 • An unconnected graph such that each • An undirected graph is a tree if and only if connectdted componen tit is a tree thihere is a un ique s ilimple pat hbh between any two of its vertices prepared by Nattee Niparnan prepared by Nattee Niparnan Rooted Tree Example • Pick any vertex and declare it as a – We can assign a direction to the tree as the direction from the root to every vertex – A directed graph result from such direction is the – Different root yields different prepared by Nattee Niparnan prepared by Nattee Niparnan Terminology for a rooted tree Terminology for a rooted tree • Parent • Definition is a parent of when there is a directed edge from u to is the child of • Any vertex having no children • Sibling – Vertices with the same parent • Ancestors of • Any vertex not being a leaf – Any vertex in the path from the root to – i.e., the parent, the parent of the parent, the parent of the parent of the parent,… • Descendant of – Any vertex having the vertex as its ancestor prepared by Nattee Niparnan prepared by Nattee Niparnan Example Subtree • Definition – A with v as its root is a subgraph of a tree that consists of v and all its descendants and the corresponding edges prepared by Nattee Niparnan prepared by Nattee Niparnan M‐ary tree Example • Definition – A tree is call if every internal vertex has no more than m children – A tree is called “ ” when every internal vertex has exactly m children – And m‐ary tree where m = 2 is called a binary tree prepared by Nattee Niparnan prepared by Nattee Niparnan An Ordered Tree Tree as a model • A rooted tree such that we have “ordered” on the c hildren o f an in terna l no de – When we draw an ordered tree, the children are drawn from left to right according to its order • For binary tree – The first child is called the “ ” child • The tree rooted at the left child is called the left subtree – The second child is called the “ ” child • The tree rooted at the right child is called the right subtree prepared by Nattee Niparnan prepared by Nattee Niparnan Tree as a model Properties of Trees • Theorem 2 – ATreewithA Tree with vertices has edges • Proof – Convert a tree into a rooted tree – we see that the arrow of the directed tree point out from root – For every directed edge, there is one vertex associated with the edge • Except the root – Since we have non‐root vertices • Then we have exactly edge prepared by Nattee Niparnan prepared by Nattee Niparnan Properties of Trees Properties of Trees • Theorem 3 • Theorem 4 – A full m‐ary tree with internal vertices – A full m‐ary tree with contains vertices 1) n vertices has internal vertices and • Proof leaves 2) i internal vertices has vertices and – Every vertex , except the root, in a tree is a child of an internal vertex 3) lleaveshasl leaves has vertices and – Since the tree is “full”, every internal vertex has internal vertices exactly children • So we have vertices, exclude the root prepared by Nattee Niparnan prepared by Nattee Niparnan Proof of theorem 4 Balanced Tree • Let define for a full m‐ary tree • It is a desirable property to have a tree that – l = number of leaves – i = number of internal vertices i“lis “almost ”fll” full – n = total number of vertices – i.e., we could say that we like balanced tree • We use theorem 3 and a fact that (because a vertex must be either a leave or an internal – Intuitively, a tree such that each subtree are of vertex similar size • Try 1) – Solve n = mi + 1 for i • i= (n –1)/m – Put this back into n = i+ l yields • l = n –i= n –(n –1)/m • = mn/m –(n­1)/m = [nm –(n –1)]/m = [(m –1)n + 1]/m prepared by Nattee Niparnan prepared by Nattee Niparnan Level and Height Example • Definition – The of a vertex is the length of the unique path from the root to • The level of the root is defined as 0 – The of a rooted tree is the maximum of the levels of its vertices prepared by Nattee Niparnan prepared by Nattee Niparnan Balanced Tree Theorem 5 • Definition • There are at most leaves in an m‐ary tree – A balanced tree is a rooted m‐ary tree of height ofhf he ig ht such that all leaves are at level h or • Proof – By induction on the height of a tree prepared by Nattee Niparnan prepared by Nattee Niparnan Proof: The basis step Proof: the Inductive Step • The base case • Assume that the result is true for the height • TfhihTree of height 1 lhhiiifhless than h, i.e., it is true for h‐1, h‐2, h‐3, … – The root is the only internal vertex but not h – We can’t have more than m children for the root • Consider a tree of height • These children are leaves – Take out the root of • So, the number of leaves is at most – Consider the remaining subtree of the root • These subtrees are of height at most • Hence, we can say that each subtree has at most • Hence, the base case is correct leaves prepared by Nattee Niparnan prepared by Nattee Niparnan Proof: the Inductive step Corollary 1 • Now, the leaves of these subtrees are exactly • If an m‐ary tree of height has leaves, then the leaves of • Since we have at most subtrees • If an m‐ary tree is full and balance, then – The number of leaves is at most – (note that ceil is the ceiling function (round up)) • Hence, the theorem is correct by the • The proof is simple, try to read it from the principle of mathematical induction book prepared by Nattee Niparnan prepared by Nattee Niparnan Decision Tree • We use tree to describe a process involving serifdiiies of decision Application of Tree Is a lecturer? yes no Section 10.2 Grant Have an ID card? access yes no Grant Deny prepared by Nattee Niparnanaccess access Eight coin problem Comparing three variables • We have 8 coins • We have a,b,c all are distinct – One o f them i s a fa ke co in, w hic h is lig hter – We whhhich to sort t hese varia ble – We have a balance scale – How to determine a fake one? prepared by Nattee Niparnan prepared by Nattee Niparnan.
Load more

Recommended publications
  • 1 Suffix Trees
    This material takes about 1.5 hours. 1 Suffix Trees Gusfield: Algorithms on Strings, Trees, and Sequences. Weiner 73 “Linear Pattern-matching algorithms” IEEE conference on automata and switching theory McCreight 76 “A space-economical suffix tree construction algorithm” JACM 23(2) 1976 Chen and Seifras 85 “Efficient and Elegegant Suffix tree construction” in Apos- tolico/Galil Combninatorial Algorithms on Words Another “search” structure, dedicated to strings. Basic problem: match a “pattern” (of length m) to “text” (of length n) • goal: decide if a given string (“pattern”) is a substring of the text • possibly created by concatenating short ones, eg newspaper • application in IR, also computational bio (DNA seqs) • if pattern avilable first, can build DFA, run in time linear in text • if text available first, can build suffix tree, run in time linear in pattern. • applications in computational bio. First idea: binary tree on strings. Inefficient because run over pattern many times. • fractional cascading? • realize only need one character at each node! Tries: • used to store dictionary of strings • trees with children indexed by “alphabet” • time to search equal length of query string • insertion ditto. • optimal, since even hashing requires this time to hash. • but better, because no “hash function” computed. • space an issue: – using array increases stroage cost by |Σ| – using binary tree on alphabet increases search time by log |Σ| 1 – ok for “const alphabet” – if really fussy, could use hash-table at each node. • size in worst case: sum of word lengths (so pretty much solves “dictionary” problem. But what about substrings? • Relevance to DNA searches • idea: trie of all n2 substrings • equivalent to trie of all n suffixes.
    [Show full text]
  • Balanced Trees Part One
    Balanced Trees Part One Balanced Trees ● Balanced search trees are among the most useful and versatile data structures. ● Many programming languages ship with a balanced tree library. ● C++: std::map / std::set ● Java: TreeMap / TreeSet ● Many advanced data structures are layered on top of balanced trees. ● We’ll see several later in the quarter! Where We're Going ● B-Trees (Today) ● A simple type of balanced tree developed for block storage. ● Red/Black Trees (Today/Thursday) ● The canonical balanced binary search tree. ● Augmented Search Trees (Thursday) ● Adding extra information to balanced trees to supercharge the data structure. Outline for Today ● BST Review ● Refresher on basic BST concepts and runtimes. ● Overview of Red/Black Trees ● What we're building toward. ● B-Trees and 2-3-4 Trees ● Simple balanced trees, in depth. ● Intuiting Red/Black Trees ● A much better feel for red/black trees. A Quick BST Review Binary Search Trees ● A binary search tree is a binary tree with 9 the following properties: 5 13 ● Each node in the BST stores a key, and 1 6 10 14 optionally, some auxiliary information. 3 7 11 15 ● The key of every node in a BST is strictly greater than all keys 2 4 8 12 to its left and strictly smaller than all keys to its right. Binary Search Trees ● The height of a binary search tree is the 9 length of the longest path from the root to a 5 13 leaf, measured in the number of edges. 1 6 10 14 ● A tree with one node has height 0.
    [Show full text]
  • Heaps a Heap Is a Complete Binary Tree. a Max-Heap Is A
    Heaps Heaps 1 A heap is a complete binary tree. A max-heap is a complete binary tree in which the value in each internal node is greater than or equal to the values in the children of that node. A min-heap is defined similarly. 97 Mapping the elements of 93 84 a heap into an array is trivial: if a node is stored at 90 79 83 81 index k, then its left child is stored at index 42 55 73 21 83 2k+1 and its right child at index 2k+2 01234567891011 97 93 84 90 79 83 81 42 55 73 21 83 CS@VT Data Structures & Algorithms ©2000-2009 McQuain Building a Heap Heaps 2 The fact that a heap is a complete binary tree allows it to be efficiently represented using a simple array. Given an array of N values, a heap containing those values can be built, in situ, by simply “sifting” each internal node down to its proper location: - start with the last 73 73 internal node * - swap the current 74 81 74 * 93 internal node with its larger child, if 79 90 93 79 90 81 necessary - then follow the swapped node down 73 * 93 - continue until all * internal nodes are 90 93 90 73 done 79 74 81 79 74 81 CS@VT Data Structures & Algorithms ©2000-2009 McQuain Heap Class Interface Heaps 3 We will consider a somewhat minimal maxheap class: public class BinaryHeap<T extends Comparable<? super T>> { private static final int DEFCAP = 10; // default array size private int size; // # elems in array private T [] elems; // array of elems public BinaryHeap() { .
    [Show full text]
  • CSCI 333 Data Structures Binary Trees Binary Tree Example Full And
    Notes with the dark blue background CSCI 333 were prepared by the textbook author Data Structures Clifford A. Shaffer Chapter 5 Department of Computer Science 18, 20, 23, and 25 September 2002 Virginia Tech Copyright © 2000, 2001 Binary Trees Binary Tree Example A binary tree is made up of a finite set of Notation: Node, nodes that is either empty or consists of a children, edge, node called the root together with two parent, ancestor, binary trees, called the left and right descendant, path, subtrees, which are disjoint from each depth, height, level, other and from the root. leaf node, internal node, subtree. Full and Complete Binary Trees Full Binary Tree Theorem (1) Full binary tree: Each node is either a leaf or Theorem: The number of leaves in a non-empty internal node with exactly two non-empty children. full binary tree is one more than the number of internal nodes. Complete binary tree: If the height of the tree is d, then all leaves except possibly level d are Proof (by Mathematical Induction): completely full. The bottom level has all nodes to the left side. Base case: A full binary tree with 1 internal node must have two leaf nodes. Induction Hypothesis: Assume any full binary tree T containing n-1 internal nodes has n leaves. 1 Full Binary Tree Theorem (2) Full Binary Tree Corollary Induction Step: Given tree T with n internal Theorem: The number of null pointers in a nodes, pick internal node I with two leaf children. non-empty tree is one more than the Remove I’s children, call resulting tree T’.
    [Show full text]
  • Tree Structures
    Tree Structures Definitions: o A tree is a connected acyclic graph. o A disconnected acyclic graph is called a forest o A tree is a connected digraph with these properties: . There is exactly one node (Root) with in-degree=0 . All other nodes have in-degree=1 . A leaf is a node with out-degree=0 . There is exactly one path from the root to any leaf o The degree of a tree is the maximum out-degree of the nodes in the tree. o If (X,Y) is a path: X is an ancestor of Y, and Y is a descendant of X. Root X Y CSci 1112 – Algorithms and Data Structures, A. Bellaachia Page 1 Level of a node: Level 0 or 1 1 or 2 2 or 3 3 or 4 Height or depth: o The depth of a node is the number of edges from the root to the node. o The root node has depth zero o The height of a node is the number of edges from the node to the deepest leaf. o The height of a tree is a height of the root. o The height of the root is the height of the tree o Leaf nodes have height zero o A tree with only a single node (hence both a root and leaf) has depth and height zero. o An empty tree (tree with no nodes) has depth and height −1. o It is the maximum level of any node in the tree. CSci 1112 – Algorithms and Data Structures, A.
    [Show full text]
  • Binary Search Tree
    ADT Binary Search Tree! Ellen Walker! CPSC 201 Data Structures! Hiram College! Binary Search Tree! •" Value-based storage of information! –" Data is stored in order! –" Data can be retrieved by value efficiently! •" Is a binary tree! –" Everything in left subtree is < root! –" Everything in right subtree is >root! –" Both left and right subtrees are also BST#s! Operations on BST! •" Some can be inherited from binary tree! –" Constructor (for empty tree)! –" Inorder, Preorder, and Postorder traversal! •" Some must be defined ! –" Insert item! –" Delete item! –" Retrieve item! The Node<E> Class! •" Just as for a linked list, a node consists of a data part and links to successor nodes! •" The data part is a reference to type E! •" A binary tree node must have links to both its left and right subtrees! The BinaryTree<E> Class! The BinaryTree<E> Class (continued)! Overview of a Binary Search Tree! •" Binary search tree definition! –" A set of nodes T is a binary search tree if either of the following is true! •" T is empty! •" Its root has two subtrees such that each is a binary search tree and the value in the root is greater than all values of the left subtree but less than all values in the right subtree! Overview of a Binary Search Tree (continued)! Searching a Binary Tree! Class TreeSet and Interface Search Tree! BinarySearchTree Class! BST Algorithms! •" Search! •" Insert! •" Delete! •" Print values in order! –" We already know this, it#s inorder traversal! –" That#s why it#s called “in order”! Searching the Binary Tree! •" If the tree is
    [Show full text]
  • Suffix Trees and Suffix Arrays in Primary and Secondary Storage Pang Ko Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2007 Suffix trees and suffix arrays in primary and secondary storage Pang Ko Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Bioinformatics Commons, and the Computer Sciences Commons Recommended Citation Ko, Pang, "Suffix trees and suffix arrays in primary and secondary storage" (2007). Retrospective Theses and Dissertations. 15942. https://lib.dr.iastate.edu/rtd/15942 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Suffix trees and suffix arrays in primary and secondary storage by Pang Ko A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Computer Engineering Program of Study Committee: Srinivas Aluru, Major Professor David Fern´andez-Baca Suraj Kothari Patrick Schnable Srikanta Tirthapura Iowa State University Ames, Iowa 2007 UMI Number: 3274885 UMI Microform 3274885 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii DEDICATION To my parents iii TABLE OF CONTENTS LISTOFTABLES ................................... v LISTOFFIGURES .................................. vi ACKNOWLEDGEMENTS. .. .. .. .. .. .. .. .. .. ... .. .. .. .. vii ABSTRACT....................................... viii CHAPTER1. INTRODUCTION . 1 1.1 SuffixArrayinMainMemory .
    [Show full text]
  • Binary Tree Fall 2017 Stony Brook University Instructor: Shebuti Rayana [email protected] Introduction to Tree
    CSE 230 Intermediate Programming in C and C++ Binary Tree Fall 2017 Stony Brook University Instructor: Shebuti Rayana [email protected] Introduction to Tree ■ Tree is a non-linear data structure which is a collection of data (Node) organized in hierarchical structure. ■ In tree data structure, every individual element is called as Node. Node stores – the actual data of that particular element and – link to next element in hierarchical structure. Tree with 11 nodes and 10 edges Shebuti Rayana (CS, Stony Brook University) 2 Tree Terminology ■ Root ■ In a tree data structure, the first node is called as Root Node. Every tree must have root node. In any tree, there must be only one root node. Root node does not have any parent. (same as head in a LinkedList). Here, A is the Root node Shebuti Rayana (CS, Stony Brook University) 3 Tree Terminology ■ Edge ■ The connecting link between any two nodes is called an Edge. In a tree with 'N' number of nodes there will be a maximum of 'N-1' number of edges. Edge is the connecting link between the two nodes Shebuti Rayana (CS, Stony Brook University) 4 Tree Terminology ■ Parent ■ The node which is predecessor of any node is called as Parent Node. The node which has branch from it to any other node is called as parent node. Parent node can also be defined as "The node which has child / children". Here, A is Parent of B and C B is Parent of D, E and F C is the Parent of G and H Shebuti Rayana (CS, Stony Brook University) 5 Tree Terminology ■ Child ■ The node which is descendant of any node is called as CHILD Node.
    [Show full text]
  • Section 11.1 Introduction to Trees Definition
    Section 11.1 Introduction to Trees Definition: A tree is a connected undirected graph with no simple circuits. : A circuit is a path of length >=1 that begins and ends a the same vertex. d d Tournament Trees A common form of tree used in everyday life is the tournament tree, used to describe the outcome of a series of games, such as a tennis tournament. Alice Antonia Alice Anita Alice Abigail Abigail Alice Amy Agnes Agnes Angela Angela Angela Audrey A Family Tree Much of the tree terminology derives from family trees. Gaea Ocean Cronus Phoebe Zeus Poseidon Demeter Pluto Leto Iapetus Persephone Apollo Atlas Prometheus Ancestor Tree An inverted family tree. Important point - it is a binary tree. Iphigenia Clytemnestra Agamemnon Leda Tyndareus Aerope Atreus Catreus Forest Graphs containing no simple circuits that are not connected, but each connected component is a tree. Theorem An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices. Rooted Trees Once a vertex of a tree has been designated as the root of the tree, it is possible to assign direction to each of the edges. Rooted Trees g a e f e c b b d a d c g f root node a internal vertex parent of g b c d e f g leaf siblings h i a b c d e f g h i h i ancestors of and a b c d e f g subtree with b as its h i root subtree with c as its root m-ary trees A rooted tree is called an m-ary tree if every internal vertex has no more than m children.
    [Show full text]
  • Binary Trees
    Binary Trees Chapter 12 Fundamentals • A binary tree is a nonlinear data structure. • A binary tree is either empty or it contains a root node and left- and right- subtrees that are also binary trees. • Applications: encryption, databases, expert systems Tree Terminology root A parent left-child right-child B C siblings leaf D E F G right-subtree left-subtree More Terminology • Consider two nodes in a tree, X and Y. • X is an ancestor of Y if X is the parent of Y, or X is the ancestor of the parent of Y. It’s RECURSIVE! • Y is a descendant of X if Y is a child of X, or Y is the descendant of a child of X. More Terminology • Consider a node Y. • The depth of a node Y is 0, if the Y is the root, or 1 + the depth of the parent of Y • The depth of a tree is the maximum depth of all its leaves. More Terminology • A full binary tree is a binary tree such that - all leaves have the same depth, and - every non-leaf node has 2 children. • A complete binary tree is a binary tree such that - every level of the tree has the maximum number of nodes possible except possibly the deepest level. - at the deepest level, the nodes are as far left as possible. Tree Examples 1 2 3 4 5 A A A A A B C B B B C D E F G D E A full binary tree is always a complete binary tree.
    [Show full text]
  • Binary Search Trees! – These Are "Nonlinear" Implementations of the ADT Table
    Data Structures Topic #8 Today’s Agenda • Continue Discussing Table Abstractions • But, this time, let’s talk about them in terms of new non-linear data structures – trees – which require that our data be organized in a hierarchical fashion Tree Introduction • Remember when we learned about tables? – We found that none of the methods for implementing tables was really adequate. – With many applications, table operations end up not being as efficient as necessary. – We found that hashing is good for retrieval, but doesn't help if our goal is also to obtain a sorted list of information. Tree Introduction • We found that the binary search also allows for fast retrieval, – but is limited to array implementations versus linked list. – Because of this, we need to move to more sophisticated implementations of tables, using binary search trees! – These are "nonlinear" implementations of the ADT table. Tree Terminology • Trees are used to represent the relationship between data items. – All trees are hierarchical in nature which means there is a parent-child relationship between "nodes" in a tree. – The lines between nodes are called directed edges. – If there is a directed edge from node A to node B -- then A is the parent of B and B is a child of A. Tree Terminology • Children of the same parent are called siblings. • Each node in a tree has at most one parent, starting at the top with the root node (which has no parent). • Parent of n The node directly above node n in the tree • Child of n The node directly below the node n in the tree Tree
    [Show full text]
  • Balanced Binary Search Trees – AVL Trees, 2-3 Trees, B-Trees
    Balanced binary search trees – AVL trees, 2‐3 trees, B‐trees Professor Clark F. Olson (with edits by Carol Zander) AVL trees One potential problem with an ordinary binary search tree is that it can have a height that is O(n), where n is the number of items stored in the tree. This occurs when the items are inserted in (nearly) sorted order. We can fix this problem if we can enforce that the tree remains balanced while still inserting and deleting items in O(log n) time. The first (and simplest) data structure to be discovered for which this could be achieved is the AVL tree, which is names after the two Russians who discovered them, Georgy Adelson‐Velskii and Yevgeniy Landis. It takes longer (on average) to insert and delete in an AVL tree, since the tree must remain balanced, but it is faster (on average) to retrieve. An AVL tree must have the following properties: • It is a binary search tree. • For each node in the tree, the height of the left subtree and the height of the right subtree differ by at most one (the balance property). The height of each node is stored in the node to facilitate determining whether this is the case. The height of an AVL tree is logarithmic in the number of nodes. This allows insert/delete/retrieve to all be performed in O(log n) time. Here is an example of an AVL tree: 18 3 37 2 11 25 40 1 8 13 42 6 10 15 Inserting 0 or 5 or 16 or 43 would result in an unbalanced tree.
    [Show full text]