CS302 - Data Structures using C++
|
|
|
- Clement Arnold
- 5 years ago
- Views:
Transcription
1 CS302 - Data Structures using C++ Topic: Trees Kostas Alexis
2 Trees List, stacks, and queues are linear in their organization of data. Items are one after another In this section, we organize data in a nonlinear hierarchical form. Item can have more than one immediate successor.
3 Terminology Use trees to represent relationships
4 Terminology Trees are hierarchical in nature Means a parent-child relationship between nodes A tree and one of its subtrees
5 Terminology Further visual examples
6 Node with no children: leaf Terminology If each node has no more than n-children then this tree is called n-ary tree Further visual examples A Root Parent of G If each node has no more than 2-children then this tree is called a binary tree Height of Tree T = 1 + height of tallest subtree of T Edge B C Node D E F G Child of C Descendant of C K H M L M and L are Siblings
7 Terminology A Length of a path = number of edges Depth of a node x = length of path from root to x Height of node x = length of longest path from x to leaf Depth and height of tree = height of root The label of a node: A, B, C B C D E F G K H M L
8 Terminology Graph-theoretic definition of a Tree: A tree is a graph for which there exists a node, called root such that For any node x, there exists exactly one path from the root to x Recursive Definition of a Tree: A tree is either Empty or It has a node called the root, followed by zero or more trees called subtrees
9 Terminology Think of a Tree in a recursive manner
10 Kinds of Trees (first examples) General Tree Set T of one or more nodes T is partitioned into disjoint subsets Binary Tree Set of T nodes either empty or partitioned into disjoint subsets Single node r, the root Two (possibly empty) sets left and right subtrees
11 Kinds of Trees Binary Trees that represent algebraic expressions
12 Kinds of Trees Binary Trees that represent algebraic expressions
13 Kinds of Trees Binary Trees that represent algebraic expressions
14 Kinds of Trees Binary Search Tree of names
15 The Height of Trees Level of a node, n If n is root, level 1 If n not the root, level is 1 greater than level of its parent Height of a tree Number of nodes on longest path from root to a leaf T empty, height 0 T not empty, height equal to max level of nodes
16 The Height of Trees Binary Trees with the same nodes but different heights
17 Full, Complete, and Balanced Binary Trees A full binary tree (sometimes also called a proper tree or a 2-tree) is a tree in which every node other than the leaves has two children. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled and all nodes are as far left as possible.
18 Full, Complete, and Balanced Binary Trees A full binary tree of height 3
19 Full, Complete, and Balanced Binary Trees A complete binary tree
20 Full, Complete, and Balanced Binary Trees A balanced binary tree is a binary tree in which the left and right subtrees of every node differ in height by no more than 1.
21 Max and Min Heights of a Binary Tree Binary Tree with n nodes Max height is n To minimize the height of a binary tree of n nodes Fill each level of a tree as completely as possible A complete tree meets this requirement
22 Max and Min Heights of a Binary Tree Binary Trees of height 3
23 Max and Min Heights of a Binary Tree Counting the nodes in a full binary tree of height h Level Number of nodes at this level Total number of nodes at this level and all previous levels 1 1 = 2 0 1= = 2 1 3= = 2 2 7= = = h 2 h-1 2 h -1...
24 Max and Min Heights of a Binary Tree Filling in the last level of a tree
25 The ADT Binary Tree Operations of ADT binary tree Add, remove Set, retrieve data Test for empty Traversal operations that visit every node Traversal can visit nodes in several different orders
26 Traversals of a Binary Tree Pseudocode for general form of a recursive traversal algoritm if (T is not empty) Display the data in T s root Traverse T s left subtree Traverse T s right subtree
27 Traversals of a Binary Tree Options for when to visit the root (defines traversal type) Preorder: before it traverses both subtrees Inorder: after it traverses left subtree, before it traverses right subtree Postorder: after it traverses both subtrees Node traversal is O(n)
28 Traversals of a Binary Tree Three traversals of a binary tree
29 Traversals of a Binary Tree Preorder traversal algorithm Process (visit) root Traverse(Left subtree of Tree s root) Traverse(Right subtree of Tree s root) Exploits recursive character of how a tree is built A type of depth-first traversal
30 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
31 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
32 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
33 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
34 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
35 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
36 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
37 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
38 Traversals of a Binary Tree Preorder traversal algorithm // Traverses the given binary tree in preorder // Assumes that visit a node means to process the node s data item preorder(bintree: BinaryTree): void if (bintree is not empty) Visit the root of bintree preorder(left subtree of bintree s root) preorder(right subtree of bintree s root)
39 Traversals of a Binary Tree Inorder traversal algorithm Traverse(Left subtree of Tree s root) Process (visit) root Traverse(Right subtree of Tree s root)
40 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
41 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
42 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
43 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
44 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
45 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
46 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
47 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
48 Traversals of a Binary Tree Inorder traversal algorithm // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
49 Traversals of a Binary Tree Postorder traversal algorithm Traverse(Left subtree of Tree s root) Traverse(Right subtree of Tree s root) Process (visit) root
50 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
51 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
52 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
53 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
54 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
55 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
56 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
57 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
58 Traversals of a Binary Tree Postorder traversal algorithm // Traverses the given binary tree in postorder // Assumes that visit a node means to process the node s data item postorder(bintree: BinaryTree): void if (bintree is not empty) postorder(left subtree of bintree s root) postorder(right subtree of bintree s root) Visit the root of bintree
59 Binary Tree Operations UML diagram for the class BinaryTree UML Notation BinaryTree +isempty(): boolean +getheight(): integer +getnumberofnodes(): integer +getrootdata(): ItemType +setrootdata(newdata: ItemType): void +add(newdata: ItemType): boolean +remove(target: ItemType): boolean +clear(): void +getentry(target: ItemType): ItemType +contains(target: ItemType): boolean +preordertraverse(visit(item: ItemType): void): void +inordertraverse(visit(item: ItemType): void): void +postordertraverse(visit(item: ItemType): void): void
60 Interface for the ADT Binary Tree An interface template for the ADT Binary Tree #ifndef BINARY_TREE_INTERFACE_ #define BINARY_TREE_INTERFACE_ #include NotFoundException.h template<class ItemType> class BinaryTreeInterface public: // Tests whether this binary tree is empty True if the binary tree is empty, or false if not virtual bool isempty() const = 0; // Gets the height of this binary tree The height of the binary tree virtual int getheight() const = 0; // Gets the number of nodes in this binary tree The number of nodes in the binary tree virtual int getnumberofnodes() const = 0; // Gets the data that is in the root of the binary tree The binary tree is not empty The root s data has been returned, binary tree is unchanged The data n the root of the binary tree virtual ItemType getrootdata() const = 0
61 Interface for the ADT Binary Tree An interface template for the ADT Binary Tree // Replaces the data in the root of this binary tree with the given data, // if the tree is not empty. However, if the tree is empty, inserts a new // root node containing the given data into the tree None The data in the root of the binary tree is as given newdata The data for the root virtual bool setrootdata(const ItemType& newdata) = 0; // Adds the given data to this binary tree newdata The data to add to the binary tree The binary tree contains the new data True if the addition is successful, or false if not virtual bool add(const ItemType& newdata) = 0; // Removes the specified data from this binary tree target The data to remove from the binary tree True if the remove is successful, or false if not virtual bool remove(const ItemType& target) = 0; // Removes all data from this binary tree virtual void clear() = 0; // Retrieves the specified data from this binary tree The desired data has been returned, and the binary tree is unchanged. // If no such data was found, exception is thrown target The data to locate The data in the binary tree that matches the given data virtual ItemType getentry(const ItemType& target) const = 0;
62 Interface for the ADT Binary Tree An interface template for the ADT Binary Tree // Tests whether the specified data occurs in this binary tree The binary tree is unchanged target The data to find True if data matching the target occurs in the tree, or false otherwise virtual bool contains(const ItemType& target) const = 0; // Traverses this binary tree in preorder (inorder, postorder) and // calls the function visit once for each node visit A client-defined function that performs an operation on either each visited node or its data virtual void preordertraverse(void visit(itemtype&)) const = 0; virtual void inordertraverse(void visit(itemtype&)) const = 0; virtual void postordertraverse(void visit(itemtype&)) const = 0; // Destroys this tree and frees its assigned memory virtual ~BinaryTreeInterface() ; // end BinaryTreeInterface #endif;
63 The ADT Binary Search Tree Recursive definition of a binary search tree The value of n is greater than all values in its left subtree T L The value of n is less than all values in its right subtree T R Both T L and T R are binary search trees
64 The ADT Binary Search Tree Notice: we distinguish between left and right child A A B C B C F F G H G H
65 The ADT Binary Search Tree Search tree property All keys in left subtree smaller than root s key 8 All keys in right subtree larger than root s key Result: 5 11 Easy to find any given key Inserts/deletes by changing links
66 The ADT Binary Search Tree A binary search tree of names
67 Binary Search Tree Operations Binary search trees with the same data
68 Binary Search Tree Operations Test whether a binary search tree is empty Get the height of a binary search tree Get the number of nodes in a binary search tree Get the data in a binary search tree s root Add the given data item to a binary search tree Remove the specified data item from a binary search tree Remove all data items from a binary search tree Retrieve the specified data item in a binary search tree Test whether a binary search tree contains specific data Traverse the nodes in a binary search tree in preorder, inorder, or postorder sense These operations define the ADT binary search tree
69 Searching a Binary Search Tree Search algorithm for a binary search tree // Searches the binary search tree for a given target value search(bsttree: BinarySearchTree, target: ItemType) if (bsttree is empty) The desired item is not found else if (target == data item in the root of bsttree) The desired item is found else if (target < data item in the root of bsttree) search(left subtree of bsttree, target) else search(right subtree of bsttree, target)
70 Creating a Binary Search Tree Empty subtree where the search algorithm terminates when looking for Finn
71 Traversals of a Binary Search Tree Inorder traversal of a binary search tree visits tree s nodes in sorted search-key order // Traverses the given binary tree in inorder // Assumes that visit a node means to process the node s data item inorder(bintree: BinaryTree): void if (bintree is not empty) inorder(left subtree of bintree s root) Visit the root of bintree inorder(right subtree of bintree s root)
72 Efficiency of Binary Search Tree Operations Max number of comparisons for retrieval, addition, or removal The height of the tree Adding entries in sorted order Produces maximum-height binary search tree Adding entries in random order Produces near-minimum-height binary search tree
73 Efficiency of Binary Search Tree Operations The Big O for the retrieval, addition, removal, and traversal operations of the ADT binary search tree (BST): Operation Average Case Worst Case Retrieval O(logn) O(n) Addition O(logn) O(n) Removal O(logn) O(n) Traversal O(n) O(n)
74 Thank you
A set of nodes (or vertices) with a single starting point
Binary Search Trees Understand tree terminology Understand and implement tree traversals Define the binary search tree property Implement binary search trees Implement the TreeSort algorithm 2 A set of
Introduction to Binary Trees
Introduction to inary Trees 1 ackground ll data structures examined so far are linear data structures. Each element in a linear data structure has a clear predecessor and a clear successor. Precessors
CS302 - Data Structures using C++
CS302 - Data Structures using C++ Topic: Implementation Kostas Alexis s Definition: A (BST) is a binary tree where each node has a Comparable key (and an associated value) and satisfies the restriction
Programming II (CS300)
1 Programming II (CS300) Chapter 11: Binary Search Trees MOUNA KACEM mouna@cs.wisc.edu Fall 2018 General Overview of Data Structures 2 Introduction to trees 3 Tree: Important non-linear data structure
Trees. Introduction & Terminology. February 05, 2018 Cinda Heeren / Geoffrey Tien 1
Trees Introduction & Terminology Cinda Heeren / Geoffrey Tien 1 Review: linked lists Linked lists are constructed out of nodes, consisting of a data element a pointer to another node Lists are constructed
CS302 - Data Structures using C++
CS302 - Data Structures using C++ Topic: Tree Implementations Kostas Alexis Nodes in a Binary Tree Representing tree nodes Must contain both data and pointers to node s children Each node will be an object
CS302 - Data Structures using C++
CS302 - Data Structures using C++ Topic: Heaps Kostas Alexis The ADT Heap A heap is a complete binary tree that is either: Empty or Whose root contains a value >= each of its children and has heaps as
3137 Data Structures and Algorithms in C++
3137 Data Structures and Algorithms in C++ Lecture 3 July 12 2006 Shlomo Hershkop 1 Announcements Homework 2 out tonight Please make sure you complete hw1 asap if you have issues, please contact me will
Chapter 20: Binary Trees
Chapter 20: Binary Trees 20.1 Definition and Application of Binary Trees Definition and Application of Binary Trees Binary tree: a nonlinear linked list in which each node may point to 0, 1, or two other
Trees. (Trees) Data Structures and Programming Spring / 28
Trees (Trees) Data Structures and Programming Spring 2018 1 / 28 Trees A tree is a collection of nodes, which can be empty (recursive definition) If not empty, a tree consists of a distinguished node r
Chapter 11.!!!!Trees! 2011 Pearson Addison-Wesley. All rights reserved 11 A-1
Chapter 11!!!!Trees! 2011 Pearson Addison-Wesley. All rights reserved 11 A-1 2015-12-01 09:30:53 1/54 Chapter-11.pdf (#13) Terminology Definition of a general tree! A general tree T is a set of one or
Chapter 11.!!!!Trees! 2011 Pearson Addison-Wesley. All rights reserved 11 A-1
Chapter 11!!!!Trees! 2011 Pearson Addison-Wesley. All rights reserved 11 A-1 2015-03-25 21:47:41 1/53 Chapter-11.pdf (#4) Terminology Definition of a general tree! A general tree T is a set of one or more
Trees. Q: Why study trees? A: Many advance ADTs are implemented using tree-based data structures.
Trees Q: Why study trees? : Many advance DTs are implemented using tree-based data structures. Recursive Definition of (Rooted) Tree: Let T be a set with n 0 elements. (i) If n = 0, T is an empty tree,
Binary Trees, Binary Search Trees
Binary Trees, Binary Search Trees Trees Linear access time of linked lists is prohibitive Does there exist any simple data structure for which the running time of most operations (search, insert, delete)
Binary Trees
Binary Trees 4-7-2005 Opening Discussion What did we talk about last class? Do you have any code to show? Do you have any questions about the assignment? What is a Tree? You are all familiar with what
TREES. Trees - Introduction
TREES Chapter 6 Trees - Introduction All previous data organizations we've studied are linear each element can have only one predecessor and successor Accessing all elements in a linear sequence is O(n)
Tree: non-recursive definition. Trees, Binary Search Trees, and Heaps. Tree: recursive definition. Tree: example.
Trees, Binary Search Trees, and Heaps CS 5301 Fall 2013 Jill Seaman Tree: non-recursive definition Tree: set of nodes and directed edges - root: one node is distinguished as the root - Every node (except
Chapter Contents. Trees. Tree Concepts. Hierarchical Organization. Hierarchical Organization. Hierarchical Organization.
Chapter Contents Chapter 18 Tree Concepts Hierarchical Organizations Tree Terminology Traversals of a Tree Traversals of a Binary Tree Traversals of a General Tree Java Interfaces for Interfaces for All
Trees 11/15/16. Chapter 11. Terminology. Terminology. Terminology. Terminology. Terminology
Chapter 11 Trees Definition of a general tree A general tree T is a set of one or more nodes such that T is partitioned into disjoint subsets: A single node r, the root Sets that are general trees, called
BINARY SEARCH TREES. Problem Solving with Computers-II
BINARY SEARCH TREES Problem Solving with Computers-II !2 Binary Search Trees What are the operations supported? What are the running times of these operations? I Connel Lectures How do you implement the
6-TREE. Tree: Directed Tree: A directed tree is an acyclic digraph which has one node called the root node
6-TREE Data Structure Management (330701) Tree: A tree is defined as a finite set of one or more nodes such that There is a special node called the root node R. The remaining nodes are divided into n 0
Terminology. The ADT Binary Tree. The ADT Binary Search Tree
Terminology The ADT Binary Tree The ADT Binary Search Tree 1 Terminology 3 A general tree A general tree T is a set of one or more nodes such that T is partitioned into disjoint subsets: o A single node
Trees! Ellen Walker! CPSC 201 Data Structures! Hiram College!
Trees! Ellen Walker! CPSC 201 Data Structures! Hiram College! ADTʼs Weʼve Studied! Position-oriented ADT! List! Stack! Queue! Value-oriented ADT! Sorted list! All of these are linear! One previous item;
Lecture Topics. Object Structures. Hierarchical Organization. Tree Concepts. Hierarchical Organization. Hierarchical Organization
Object Structures Trees CISH 4020 Tom Blough blought@rh.edu www.rh.edu/~blought 860.618.4148 Lecture Topics Tree Concepts Traversals of a Tree Java Interfaces for Trees Examples of Binary Trees Examples
Programming II (CS300)
1 Programming II (CS300) Chapter 10: Search and Heaps MOUNA KACEM mouna@cs.wisc.edu Spring 2018 Search and Heaps 2 Linear Search Binary Search Introduction to trees Priority Queues Heaps Linear Search
Trees, Binary Trees, and Binary Search Trees
COMP171 Trees, Binary Trees, and Binary Search Trees 2 Trees Linear access time of linked lists is prohibitive Does there exist any simple data structure for which the running time of most operations (search,
Trees. Tree Structure Binary Tree Tree Traversals
Trees Tree Structure Binary Tree Tree Traversals The Tree Structure Consists of nodes and edges that organize data in a hierarchical fashion. nodes store the data elements. edges connect the nodes. The
BBM 201 Data structures
BBM 201 Data structures Lecture 11: Trees 2018-2019 Fall Content Terminology The Binary Tree The Binary Search Tree Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, 2013
Topic 14. The BinaryTree ADT
Topic 14 The BinaryTree ADT Objectives Define trees as data structures Define the terms associated with trees Discuss tree traversal algorithms Discuss a binary tree implementation Examine a binary tree
CSCI-401 Examlet #5. Name: Class: Date: True/False Indicate whether the sentence or statement is true or false.
Name: Class: Date: CSCI-401 Examlet #5 True/False Indicate whether the sentence or statement is true or false. 1. The root node of the standard binary tree can be drawn anywhere in the tree diagram. 2.
CSE 230 Intermediate Programming in C and C++ Binary Tree
CSE 230 Intermediate Programming in C and C++ Binary Tree Fall 2017 Stony Brook University Instructor: Shebuti Rayana shebuti.rayana@stonybrook.edu Introduction to Tree Tree is a non-linear data structure
Data Structure Lecture#10: Binary Trees (Chapter 5) U Kang Seoul National University
Data Structure Lecture#10: Binary Trees (Chapter 5) U Kang Seoul National University U Kang (2016) 1 In This Lecture The concept of binary tree, its terms, and its operations Full binary tree theorem Idea
Algorithms and Data Structures
Lesson 3: trees and visits Luciano Bononi http://www.cs.unibo.it/~bononi/ (slide credits: these slides are a revised version of slides created by Dr. Gabriele D Angelo) International
INF2220: algorithms and data structures Series 1
Universitetet i Oslo Institutt for Informatikk A. Maus, R.K. Runde, I. Yu INF2220: algorithms and data structures Series 1 Topic Trees & estimation of running time (Exercises with hints for solution) Issued:
CS24 Week 8 Lecture 1
CS24 Week 8 Lecture 1 Kyle Dewey Overview Tree terminology Tree traversals Implementation (if time) Terminology Node The most basic component of a tree - the squares Edge The connections between nodes
! Tree: set of nodes and directed edges. ! Parent: source node of directed edge. ! Child: terminal node of directed edge
Trees (& Heaps) Week 12 Gaddis: 20 Weiss: 21.1-3 CS 5301 Spring 2015 Jill Seaman 1 Tree: non-recursive definition! Tree: set of nodes and directed edges - root: one node is distinguished as the root -
Tree Terminology. root. Edge. interior node. parent. path. subtree. child. leaf
Tree Terminology Binary Trees CS00: Trees interior node Node path root parent Edge subtree Degree? Depth/Level? A binary tree is a set T of nodes such that either T is empty, or T is partitioned into three
Tree. A path is a connected sequence of edges. A tree topology is acyclic there is no loop.
Tree A tree consists of a set of nodes and a set of edges connecting pairs of nodes. A tree has the property that there is exactly one path (no more, no less) between any pair of nodes. A path is a connected
Abstract Data Structures IB Computer Science. Content developed by Dartford Grammar School Computer Science Department
Abstract Data Structures IB Computer Science Content developed by Dartford Grammar School Computer Science Department HL Topics 1-7, D1-4 1: System design 2: Computer Organisation 3: Networks 4: Computational
CSC148 Week 6. Larry Zhang
CSC148 Week 6 Larry Zhang 1 Announcements Test 1 coverage: trees (topic of today and Wednesday) are not covered Assignment 1 slides posted on the course website. 2 Data Structures 3 Data Structures A data
Data Structures. Trees. By Dr. Mohammad Ali H. Eljinini. M.A. Eljinini, PhD
Data Structures Trees By Dr. Mohammad Ali H. Eljinini Trees Are collections of items arranged in a tree like data structure (none linear). Items are stored inside units called nodes. However: We can use
Upcoming ACM Events Linux Crash Course Date: Time: Location: Weekly Crack the Coding Interview Date:
Upcoming ACM Events Linux Crash Course Date: Oct. 2nd and 3rd Time: 1:15 pm - 3:15 pm Location: UW1-210 (10/02) and UW1-221 (10/03) Weekly Crack the Coding Interview Date: Weekly Fridays from Oct. 5th
Trees. CSE 373 Data Structures
Trees CSE 373 Data Structures Readings Reading Chapter 7 Trees 2 Why Do We Need Trees? Lists, Stacks, and Queues are linear relationships Information often contains hierarchical relationships File directories
BE ISE 3 rd Semester
Scheme of valuvation 1 What is a Binary tree? State its properties? How it is represented using array and linked list, give example. binary tree is a tree data structure in which each node has at most
Name CPTR246 Spring '17 (100 total points) Exam 3
Name CPTR246 Spring '17 (100 total points) Exam 3 1. Linked Lists Consider the following linked list of integers (sorted from lowest to highest) and the changes described. Make the necessary changes in
Tree Structures. A hierarchical data structure whose point of entry is the root node
Binary Trees 1 Tree Structures A tree is A hierarchical data structure whose point of entry is the root node This structure can be partitioned into disjoint subsets These subsets are themselves trees and
Principles of Computer Science
Principles of Computer Science Binary Trees 08/11/2013 CSCI 2010 - Binary Trees - F.Z. Qureshi 1 Today s Topics Extending LinkedList with Fast Search Sorted Binary Trees Tree Concepts Traversals of a Binary
Binary Search Trees Treesort
Treesort CS 311 Data Structures and Algorithms Lecture Slides Friday, November 13, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks CHAPPELLG@member.ams.org 2005 2009
BST Implementation. Data Structures. Lecture 4 Binary search trees (BST) Dr. Mahmoud Attia Sakr University of Ain Shams
Lecture 4 Binary search trees (BST) Dr. Mahmoud Attia Sakr mahmoud.sakr@cis.asu.edu.eg Cairo, Egypt, October 2012 Binary Search Trees (BST) 1. Hierarchical data structure with a single reference to root
INF2220: algorithms and data structures Series 1
Universitetet i Oslo Institutt for Informatikk I. Yu, D. Karabeg INF2220: algorithms and data structures Series 1 Topic Function growth & estimation of running time, trees Issued: 24. 08. 2016 Exercise
Trees Chapter 19, 20. Instructor: Scott Kristjanson CMPT 125/125 SFU Burnaby, Fall 2013
Trees Chapter 19, 20 Instructor: Scott Kristjanson CMPT 125/125 SFU Burnaby, Fall 2013 2 Scope Trees: Trees as data structures Tree terminology Tree implementations Analyzing tree efficiency Tree traversals
Trees : Part 1. Section 4.1. Theory and Terminology. A Tree? A Tree? Theory and Terminology. Theory and Terminology
Trees : Part Section. () (2) Preorder, Postorder and Levelorder Traversals Definition: A tree is a connected graph with no cycles Consequences: Between any two vertices, there is exactly one unique path
a graph is a data structure made up of nodes in graph theory the links are normally called edges
1 Trees Graphs a graph is a data structure made up of nodes each node stores data each node has links to zero or more nodes in graph theory the links are normally called edges graphs occur frequently in
! Tree: set of nodes and directed edges. ! Parent: source node of directed edge. ! Child: terminal node of directed edge
Trees & Heaps Week 12 Gaddis: 20 Weiss: 21.1-3 CS 5301 Fall 2018 Jill Seaman!1 Tree: non-recursive definition! Tree: set of nodes and directed edges - root: one node is distinguished as the root - Every
MULTIMEDIA COLLEGE JALAN GURNEY KIRI KUALA LUMPUR
STUDENT IDENTIFICATION NO MULTIMEDIA COLLEGE JALAN GURNEY KIRI 54100 KUALA LUMPUR FIFTH SEMESTER FINAL EXAMINATION, 2014/2015 SESSION PSD2023 ALGORITHM & DATA STRUCTURE DSEW-E-F-2/13 25 MAY 2015 9.00 AM
Programming II (CS300)
1 Programming II (CS300) Chapter 12: Heaps and Priority Queues MOUNA KACEM mouna@cs.wisc.edu Fall 2018 Heaps and Priority Queues 2 Priority Queues Heaps Priority Queue 3 QueueADT Objects are added and
tree nonlinear Examples
The Tree ADT Objectives Define trees as data structures Define the terms associated with trees Discuss tree traversal algorithms Discuss a binary tree implementation Examine a binary tree example 10-2
Binary Trees and Binary Search Trees
Binary Trees and Binary Search Trees Learning Goals After this unit, you should be able to... Determine if a given tree is an instance of a particular type (e.g. binary, and later heap, etc.) Describe
CS 231 Data Structures and Algorithms Fall Recursion and Binary Trees Lecture 21 October 24, Prof. Zadia Codabux
CS 231 Data Structures and Algorithms Fall 2018 Recursion and Binary Trees Lecture 21 October 24, 2018 Prof. Zadia Codabux 1 Agenda ArrayQueue.java Recursion Binary Tree Terminologies Traversal 2 Administrative
Uses for Trees About Trees Binary Trees. Trees. Seth Long. January 31, 2010
Uses for About Binary January 31, 2010 Uses for About Binary Uses for Uses for About Basic Idea Implementing Binary Example: Expression Binary Search Uses for Uses for About Binary Uses for Storage Binary
Data Structures and Algorithms for Engineers
04-630 Data Structures and Algorithms for Engineers David Vernon Carnegie Mellon University Africa vernon@cmu.edu www.vernon.eu Data Structures and Algorithms for Engineers 1 Carnegie Mellon University
CS301 - Data Structures Glossary By
CS301 - Data Structures Glossary By Abstract Data Type : A set of data values and associated operations that are precisely specified independent of any particular implementation. Also known as ADT Algorithm
COMP : Trees. COMP20012 Trees 219
COMP20012 3: Trees COMP20012 Trees 219 Trees Seen lots of examples. Parse Trees Decision Trees Search Trees Family Trees Hierarchical Structures Management Directories COMP20012 Trees 220 Trees have natural
Discussion 2C Notes (Week 8, February 25) TA: Brian Choi Section Webpage:
Discussion 2C Notes (Week 8, February 25) TA: Brian Choi (schoi@cs.ucla.edu) Section Webpage: http://www.cs.ucla.edu/~schoi/cs32 Trees Definitions Yet another data structure -- trees. Just like a linked
Trees. Truong Tuan Anh CSE-HCMUT
Trees Truong Tuan Anh CSE-HCMUT Outline Basic concepts Trees Trees A tree consists of a finite set of elements, called nodes, and a finite set of directed lines, called branches, that connect the nodes
Section 5.5. Left subtree The left subtree of a vertex V on a binary tree is the graph formed by the left child L of V, the descendents
Section 5.5 Binary Tree A binary tree is a rooted tree in which each vertex has at most two children and each child is designated as being a left child or a right child. Thus, in a binary tree, each vertex
Outline. Preliminaries. Binary Trees Binary Search Trees. What is Tree? Implementation of Trees using C++ Tree traversals and applications
Trees 1 Outline Preliminaries What is Tree? Implementation of Trees using C++ Tree traversals and applications Binary Trees Binary Search Trees Structure and operations Analysis 2 What is a Tree? A tree
CISC 235 Topic 3. General Trees, Binary Trees, Binary Search Trees
CISC 235 Topic 3 General Trees, Binary Trees, Binary Search Trees Outline General Trees Terminology, Representation, Properties Binary Trees Representations, Properties, Traversals Recursive Algorithms
Friday Four Square! 4:15PM, Outside Gates
Binary Search Trees Friday Four Square! 4:15PM, Outside Gates Implementing Set On Monday and Wednesday, we saw how to implement the Map and Lexicon, respectively. Let's now turn our attention to the Set.
Computer Science 210 Data Structures Siena College Fall Topic Notes: Trees
Computer Science 0 Data Structures Siena College Fall 08 Topic Notes: Trees We ve spent a lot of time looking at a variety of structures where there is a natural linear ordering of the elements in arrays,
Data Structures and Algorithms
Data Structures and Algorithms CS245-2017S-06 Binary Search Trees David Galles Department of Computer Science University of San Francisco 06-0: Ordered List ADT Operations: Insert an element in the list
Trees. Dr. Ronaldo Menezes Hugo Serrano Ronaldo Menezes, Florida Tech
Trees Dr. Ronaldo Menezes Hugo Serrano (hbarbosafilh2011@my.fit.edu) Introduction to Trees Trees are very common in computer science They come in different variations They are used as data representation
Cpt S 122 Data Structures. Data Structures Trees
Cpt S 122 Data Structures Data Structures Trees Nirmalya Roy School of Electrical Engineering and Computer Science Washington State University Motivation Trees are one of the most important and extensively
Tree Structures. Definitions: o A tree is a connected acyclic graph. o A disconnected acyclic graph is called a forest
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)
CSC148-Section:L0301
Slides adapted from Professor Danny Heap course material winter17 CSC148-Section:L0301 Week#8-Friday Instructed by AbdulAziz Al-Helali a.alhelali@mail.utoronto.ca Office hours: Wednesday 11-1, BA2230.
TREES Lecture 10 CS2110 Spring2014
TREES Lecture 10 CS2110 Spring2014 Readings and Homework 2 Textbook, Chapter 23, 24 Homework: A thought problem (draw pictures!) Suppose you use trees to represent student schedules. For each student there
Binary Trees. Height 1
Binary Trees Definitions A tree is a finite set of one or more nodes that shows parent-child relationship such that There is a special node called root Remaining nodes are portioned into subsets T1,T2,T3.
CS 206 Introduction to Computer Science II
CS 206 Introduction to Computer Science II 10 / 10 / 2016 Instructor: Michael Eckmann Today s Topics Questions? Comments? A few comments about Doubly Linked Lists w/ dummy head/tail Trees Binary trees
SCJ2013 Data Structure & Algorithms. Binary Search Tree. Nor Bahiah Hj Ahmad
SCJ2013 Data Structure & Algorithms Binary Search Tree Nor Bahiah Hj Ahmad Binary Search Tree A binary search tree has the following properties: For every node n in the tree Value of n is greater than
If you took your exam home last time, I will still regrade it if you want.
Some Comments about HW2: 1. You should have used a generic node in your structure one that expected an Object, and not some other type. 2. Main is still too long for some people 3. braces in wrong place,
4. Trees. 4.1 Preliminaries. 4.2 Binary trees. 4.3 Binary search trees. 4.4 AVL trees. 4.5 Splay trees. 4.6 B-trees. 4. Trees
4. Trees 4.1 Preliminaries 4.2 Binary trees 4.3 Binary search trees 4.4 AVL trees 4.5 Splay trees 4.6 B-trees Malek Mouhoub, CS340 Fall 2002 1 4.1 Preliminaries A Root B C D E F G Height=3 Leaves H I J
Lec 17 April 8. Topics: binary Trees expression trees. (Chapter 5 of text)
Lec 17 April 8 Topics: binary Trees expression trees Binary Search Trees (Chapter 5 of text) Trees Linear access time of linked lists is prohibitive Heap can t support search in O(log N) time. (takes O(N)
References and Homework ABSTRACT DATA TYPES; LISTS & TREES. Abstract Data Type (ADT) 9/24/14. ADT example: Set (bunch of different values)
9// References and Homework Text: Chapters, and ABSTRACT DATA TYPES; LISTS & TREES Homework: Learn these List methods, from http://docs.oracle.com/javase/7/docs/api/java/util/list.html add, addall, contains,
Trees. Eric McCreath
Trees Eric McCreath 2 Overview In this lecture we will explore: general trees, binary trees, binary search trees, and AVL and B-Trees. 3 Trees Trees are recursive data structures. They are useful for:
CS 171: Introduction to Computer Science II. Binary Search Trees
CS 171: Introduction to Computer Science II Binary Search Trees Binary Search Trees Symbol table applications BST definitions and terminologies Search and insert Traversal Ordered operations Delete Symbol
Binary Trees Fall 2018 Margaret Reid-Miller
Binary Trees 15-121 Fall 2018 Margaret Reid-Miller Trees Fall 2018 15-121 (Reid-Miller) 2 Binary Trees A binary tree is either empty or it contains a root node and left- and right-subtrees that are also
Trees. A tree is a directed graph with the property
2: Trees Trees A tree is a directed graph with the property There is one node (the root) from which all other nodes can be reached by exactly one path. Seen lots of examples. Parse Trees Decision Trees
March 20/2003 Jayakanth Srinivasan,
Definition : A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Definition : In a multigraph G = (V, E) two or
MID TERM MEGA FILE SOLVED BY VU HELPER Which one of the following statement is NOT correct.
MID TERM MEGA FILE SOLVED BY VU HELPER Which one of the following statement is NOT correct. In linked list the elements are necessarily to be contiguous In linked list the elements may locate at far positions
DATA STRUCTURES USING C
DATA STRUCTURES USING C The British Constitution Crown Church of England Cabinet House of Commons House of Lords Supreme Court Ministers County Metropolitan Council police Rural District County Borough
Topic Binary Trees (Non-Linear Data Structures)
Topic Binary Trees (Non-Linear Data Structures) CIS210 1 Linear Data Structures Arrays Linked lists Skip lists Self-organizing lists CIS210 2 Non-Linear Data Structures Hierarchical representation? Trees
Trees. Estruturas de Dados / Programação 2 Árvores. Márcio Ribeiro twitter.com/marciomribeiro. Introduc)on. Hierarchical structure
Introduc)on Linear structures Removing this idea, treasure of applicaons Estruturas de Dados / Programação 2 Árvores Márcio Ribeiro marcio@ic.ufal.br twitter.com/marciomribeiro Hierarchical structure Companies
Chapter 5. Binary Trees
Chapter 5 Binary Trees Definitions and Properties A binary tree is made up of a finite set of elements called nodes It consists of a root and two subtrees There is an edge from the root to its children
CS 151. Binary Trees. Friday, October 5, 12
CS 151 Binary Trees 1 Binary Tree Examples Without telling you what a binary tree is, here are some examples (that I will draw on the board): The dots/circles are called nodes, or vertices (singular: one
BINARY SEARCH TREES (CONTD) Problem Solving with Computers-II
BINARY SEARCH TREES (CONTD) Problem Solving with Computers-II A node in a BST class BSTNode { public: BSTNode* left; BSTNode* right; BSTNode* parent; int const data; g mroaa struct Card char suit BSTNode(
The tree data structure. Trees COL 106. Amit Kumar Shweta Agrawal. Acknowledgement :Many slides are courtesy Douglas Harder, UWaterloo
The tree data structure 1 Trees COL 106 Amit Kumar Shweta Agrawal Acknowledgement :Many slides are courtesy Douglas Harder, UWaterloo 1 Trees The tree data structure 3 A rooted tree data structure stores
Binary Trees. For example: Jargon: General Binary Trees. root node. level: internal node. edge. leaf node. Data Structures & File Management
Binary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from
DATA STRUCTURES AND ALGORITHMS. Hierarchical data structures: AVL tree, Bayer tree, Heap
DATA STRUCTURES AND ALGORITHMS Hierarchical data structures: AVL tree, Bayer tree, Heap Summary of the previous lecture TREE is hierarchical (non linear) data structure Binary trees Definitions Full tree,
Why Do We Need Trees?
CSE 373 Lecture 6: Trees Today s agenda: Trees: Definition and terminology Traversing trees Binary search trees Inserting into and deleting from trees Covered in Chapter 4 of the text Why Do We Need Trees?
Binary Tree Node Relationships. Binary Trees. Quick Application: Expression Trees. Traversals
Binary Trees 1 Binary Tree Node Relationships 2 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the