Trees : Part 1. Section 4.1. Theory and Terminology. A Tree? A Tree? Theory and Terminology. Theory and Terminology

Size: px
Start display at page:

Download "Trees : Part 1. Section 4.1. Theory and Terminology. A Tree? A Tree? Theory and Terminology. Theory and Terminology"

Transcription

1 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 Tree? A Tree? Definition: A ed tree is a graph G such that: G is connected G has no cycles G has exactly one vertex called the of the tree Consequences The depth of a vertex v is the length of the unique path from to v G can be arranged so that the is at the top, its neighboring vertices are vertices of depth, and so on The set of all vertices of depth k is called level k of the tree

2 A Rooted Tree Rooted Tree: Recursive definition depth = height = 3 height = 2 A graph with N nodes and N - edges Graph has one r Zero or more non-empty sub-trees, each of whose is connected to r by an edge. height = height = 0 Every node except the has one parent Definition: A descending path in a ed tree is a path, whose edges go from a vertex to a deeper vertex Consequences: A unique path from the to any vertex is a descending path The length of this path is the depth of the vertex depth = depth = Definition: If there is a descending path from v to v 2, v is an ancestor of v 2, and v 2 is a descendant of v. Suppose v is a vertex of depth k: Any vertex that is adjacent to v must have depth k - or k +. depth =

3 Suppose v is a vertex of depth k: Vertices adjacent to v of depth k + are called children of v. Suppose v is a vertex of depth k: If k > 0, there is exactly one vertex of depth k that is adjacent to v in the graph. This vertex is called the parent of v. depth = depth = Definitions A vertex with no children is called a leaf Definitions Depth of a vertex v is its distance from the. Height of a vertex v is the distance of the longest path from v to one of its descendant leaves. The height of a tree is the maximum depth of its vertices depth = height depth = Definitions The is the only vertex of depth 0. The has no parent. Example of ed tree Which are the parent nodes? Which are the child nodes? Which are the leaves? What is the height and depth of the tree? What is the height and depth of node E? Node F? depth =

4 Overview of Tree Implementation Each node points to Its first child Its next sibling Back to its parent (optional) What could be an alternate representation? Tree Traversals Definition: A traversal is the process for visiting all of the vertices in a tree Often defined recursively Each kind corresponds to an iterator type Iterators are implemented non-recursively Preorder Traversal Visit vertex, then visit child vertices (recursive definition) Depth-first search Begin at Visit vertex on arrival Implementation may be recursive, stackbased, or nested loop Preorder Traversal Preorder Traversal of UNIX Directory Tree Postorder Traversal Visit child vertices, then visit vertex (recursive definition) Depth-first search Begin at Visit vertex on departure Implementation may be recursive, stackbased, or nested loop

5 Postorder Traversal Postorder Traversal Calculating Size of Directory 7 Levelorder Traversal Visit all vertices in level, starting with level 0 and increasing Breadth-first search Begin at Visit vertex on departure Only practical implementation is queuebased Levelorder Traversal 7 Tree Traversals Preorder: depth-first search (possibly stack-based), visit on arrival Postorder: depth-first search (possibly stack-based), visit on departure Levelorder: breadth-first search (queuebased), visit on departure 5

Trees. Q: Why study trees? A: Many advance ADTs are implemented using tree-based data structures.

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,

More information

Binary Trees

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

More information

Uses for Trees About Trees Binary Trees. Trees. Seth Long. January 31, 2010

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

More information

March 20/2003 Jayakanth Srinivasan,

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

More information

CSE 230 Intermediate Programming in C and C++ Binary Tree

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

More information

Trees. Truong Tuan Anh CSE-HCMUT

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

More information

Trees. (Trees) Data Structures and Programming Spring / 28

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

More information

Binary Trees, Binary Search Trees

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)

More information

Trees. CSE 373 Data Structures

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

More information

CE 221 Data Structures and Algorithms

CE 221 Data Structures and Algorithms CE 221 Data Structures and Algorithms Chapter 4: Trees (Binary) Text: Read Weiss, 4.1 4.2 Izmir University of Economics 1 Preliminaries - I (Recursive) Definition: A tree is a collection of nodes. The

More information

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected.

Chapter 4 Trees. Theorem A graph G has a spanning tree if and only if G is connected. Chapter 4 Trees 4-1 Trees and Spanning Trees Trees, T: A simple, cycle-free, loop-free graph satisfies: If v and w are vertices in T, there is a unique simple path from v to w. Eg. Trees. Spanning trees:

More information

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. 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

More information

Trees. Tree Structure Binary Tree Tree Traversals

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

More information

COSC 2011 Section N. Trees: Terminology and Basic Properties

COSC 2011 Section N. Trees: Terminology and Basic Properties COSC 2011 Tuesday, March 27 2001 Overview Trees and Binary Trees Quick review of definitions and examples Tree Algorithms Depth, Height Tree and Binary Tree Traversals Preorder, postorder, inorder Binary

More information

Topic 18 Binary Trees "A tree may grow a thousand feet tall, but its leaves will return to its roots." -Chinese Proverb

Topic 18 Binary Trees A tree may grow a thousand feet tall, but its leaves will return to its roots. -Chinese Proverb Topic 18 "A tree may grow a thousand feet tall, but its leaves will return to its roots." -Chinese Proverb Definitions A tree is an abstract data type one entry point, the root Each node is either a leaf

More information

CSE 214 Computer Science II Introduction to Tree

CSE 214 Computer Science II Introduction to Tree CSE 214 Computer Science II Introduction to Tree Fall 2017 Stony Brook University Instructor: Shebuti Rayana shebuti.rayana@stonybrook.edu http://www3.cs.stonybrook.edu/~cse214/sec02/ Tree Tree is a non-linear

More information

LECTURE 11 TREE TRAVERSALS

LECTURE 11 TREE TRAVERSALS DATA STRUCTURES AND ALGORITHMS LECTURE 11 TREE TRAVERSALS IMRAN IHSAN ASSISTANT PROFESSOR AIR UNIVERSITY, ISLAMABAD BACKGROUND All the objects stored in an array or linked list can be accessed sequentially

More information

Section Summary. Introduction to Trees Rooted Trees Trees as Models Properties of Trees

Section Summary. Introduction to Trees Rooted Trees Trees as Models Properties of Trees Chapter 11 Copyright McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Introduction to Trees Applications

More information

CSCI-401 Examlet #5. Name: Class: Date: True/False Indicate whether the sentence or statement is true or false.

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.

More information

Tree. A path is a connected sequence of edges. A tree topology is acyclic there is no loop.

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

More information

Lec 17 April 8. Topics: binary Trees expression trees. (Chapter 5 of text)

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)

More information

12/5/17. trees. CS 220: Discrete Structures and their Applications. Trees Chapter 11 in zybooks. rooted trees. rooted trees

12/5/17. trees. CS 220: Discrete Structures and their Applications. Trees Chapter 11 in zybooks. rooted trees. rooted trees trees CS 220: Discrete Structures and their Applications A tree is an undirected graph that is connected and has no cycles. Trees Chapter 11 in zybooks rooted trees Rooted trees. Given a tree T, choose

More information

Chapter Summary. Introduction to Trees Applications of Trees Tree Traversal Spanning Trees Minimum Spanning Trees

Chapter Summary. Introduction to Trees Applications of Trees Tree Traversal Spanning Trees Minimum Spanning Trees Trees Chapter 11 Chapter Summary Introduction to Trees Applications of Trees Tree Traversal Spanning Trees Minimum Spanning Trees Introduction to Trees Section 11.1 Section Summary Introduction to Trees

More information

CSC148 Week 6. Larry Zhang

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

More information

UNIT IV -NON-LINEAR DATA STRUCTURES 4.1 Trees TREE: A tree is a finite set of one or more nodes such that there is a specially designated node called the Root, and zero or more non empty sub trees T1,

More information

Programming II (CS300)

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

More information

Trees Algorhyme by Radia Perlman

Trees Algorhyme by Radia Perlman Algorhyme by Radia Perlman I think that I shall never see A graph more lovely than a tree. A tree whose crucial property Is loop-free connectivity. A tree which must be sure to span. So packets can reach

More information

7.1 Introduction. A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w

7.1 Introduction. A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w Chapter 7 Trees 7.1 Introduction A (free) tree T is A simple graph such that for every pair of vertices v and w there is a unique path from v to w Tree Terminology Parent Ancestor Child Descendant Siblings

More information

Algorithms and Data Structures

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

More information

Trees. Trees. CSE 2011 Winter 2007

Trees. Trees. CSE 2011 Winter 2007 Trees CSE 2011 Winter 2007 2/5/2007 10:00 PM 1 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,

More information

Why Do We Need Trees?

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?

More information

TREES. Trees - Introduction

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)

More information

Data Structures Question Bank Multiple Choice

Data Structures Question Bank Multiple Choice Section 1. Fundamentals: Complexity, Algorthm Analysis 1. An algorithm solves A single problem or function Multiple problems or functions Has a single programming language implementation 2. A solution

More information

Tree Structures. A hierarchical data structure whose point of entry is the root node

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

More information

Trees! Ellen Walker! CPSC 201 Data Structures! Hiram College!

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;

More information

Trees. Eric McCreath

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:

More information

Trees. Introduction & Terminology. February 05, 2018 Cinda Heeren / Geoffrey Tien 1

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

More information

Chapter 10: Trees. A tree is a connected simple undirected graph with no simple circuits.

Chapter 10: Trees. A tree is a connected simple undirected graph with no simple circuits. Chapter 10: Trees A tree is a connected simple undirected graph with no simple circuits. Properties: o There is a unique simple path between any 2 of its vertices. o No loops. o No multiple edges. Example

More information

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 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

More information

CS 151. Binary Trees. Friday, October 5, 12

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

More information

Figure 1. A breadth-first traversal.

Figure 1. A breadth-first traversal. 4.3 Tree Traversals Stepping, or iterating, through the entries of a linearly ordered list has only two obvious orders: from front to back or from back to front. There is no obvious traversal of a general

More information

Todays Lecture. Assignment 2 deadline: You have 5 Calendar days to complete.

Todays Lecture. Assignment 2 deadline: You have 5 Calendar days to complete. Trees! Todays Lecture Assignment 2 deadline: 11 pm Monday 2/17 President s day. You have 5 Calendar days to complete. Trees What are trees in computer science? Data Structures for Trees Algorithms useful

More information

birds fly S NP N VP V Graphs and trees

birds fly S NP N VP V Graphs and trees birds fly S NP VP N birds V fly S NP NP N VP V VP S NP VP birds a fly b ab = string S A B a ab b S A B A a B b S NP VP birds a fly b ab = string Grammar 1: Grammar 2: A a A a A a B A B a B b A B A b Grammar

More information

tree nonlinear Examples

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

More information

Binary Trees. Height 1

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.

More information

9/29/2016. Chapter 4 Trees. Introduction. Terminology. Terminology. Terminology. Terminology

9/29/2016. Chapter 4 Trees. Introduction. Terminology. Terminology. Terminology. Terminology Introduction Chapter 4 Trees for large input, even linear access time may be prohibitive we need data structures that exhibit average running times closer to O(log N) binary search tree 2 Terminology recursive

More information

Binary Trees Fall 2018 Margaret Reid-Miller

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

More information

Introduction to Computers and Programming. Concept Question

Introduction to Computers and Programming. Concept Question Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture 7 April 2 2004 Concept Question G1(V1,E1) A graph G(V, where E) is V1 a finite = {}, nonempty E1 = {} set of G2(V2,E2) vertices and

More information

Topic 14. The BinaryTree ADT

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

More information

Introduction. for large input, even access time may be prohibitive we need data structures that exhibit times closer to O(log N) binary search tree

Introduction. for large input, even access time may be prohibitive we need data structures that exhibit times closer to O(log N) binary search tree Chapter 4 Trees 2 Introduction for large input, even access time may be prohibitive we need data structures that exhibit running times closer to O(log N) binary search tree 3 Terminology recursive definition

More information

Algorithms. Deleting from Red-Black Trees B-Trees

Algorithms. Deleting from Red-Black Trees B-Trees Algorithms Deleting from Red-Black Trees B-Trees Recall the rules for BST deletion 1. If vertex to be deleted is a leaf, just delete it. 2. If vertex to be deleted has just one child, replace it with that

More information

Trees 11/15/16. Chapter 11. Terminology. Terminology. Terminology. Terminology. Terminology

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

More information

Discussion 2C Notes (Week 8, February 25) TA: Brian Choi Section Webpage:

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

More information

This course is intended for 3rd and/or 4th year undergraduate majors in Computer Science.

This course is intended for 3rd and/or 4th year undergraduate majors in Computer Science. Lecture 9 Graphs This course is intended for 3rd and/or 4th year undergraduate majors in Computer Science. You need to be familiar with the design and use of basic data structures such as Lists, Stacks,

More information

Trees and Tree Traversals. Binary Trees. COMP 210: Object-Oriented Programming Lecture Notes 8. Based on notes by Logan Mayfield

Trees and Tree Traversals. Binary Trees. COMP 210: Object-Oriented Programming Lecture Notes 8. Based on notes by Logan Mayfield OMP 210: Object-Oriented Programming Lecture Notes 8 Trees and Tree Traversals ased on notes by Logan Mayfield In these notes we look at inary Trees and how to traverse them. inary Trees Imagine a list.

More information

LECTURE 17 GRAPH TRAVERSALS

LECTURE 17 GRAPH TRAVERSALS DATA STRUCTURES AND ALGORITHMS LECTURE 17 GRAPH TRAVERSALS IMRAN IHSAN ASSISTANT PROFESSOR AIR UNIVERSITY, ISLAMABAD STRATEGIES Traversals of graphs are also called searches We can use either breadth-first

More information

Computer Science 210 Data Structures Siena College Fall Topic Notes: Trees

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,

More information

Graph Algorithms Using Depth First Search

Graph Algorithms Using Depth First Search Graph Algorithms Using Depth First Search Analysis of Algorithms Week 8, Lecture 1 Prepared by John Reif, Ph.D. Distinguished Professor of Computer Science Duke University Graph Algorithms Using Depth

More information

(2,4) Trees. 2/22/2006 (2,4) Trees 1

(2,4) Trees. 2/22/2006 (2,4) Trees 1 (2,4) Trees 9 2 5 7 10 14 2/22/2006 (2,4) Trees 1 Outline and Reading Multi-way search tree ( 10.4.1) Definition Search (2,4) tree ( 10.4.2) Definition Search Insertion Deletion Comparison of dictionary

More information

Binary Trees. Reading: Lewis & Chase 12.1, 12.3 Eck Programming Course CL I

Binary Trees. Reading: Lewis & Chase 12.1, 12.3 Eck Programming Course CL I inary Trees Reading: Lewis & hase 12.1, 12.3 Eck 9.4.1 Objectives Tree basics Learn the terminology used when talking about trees iscuss methods for traversing trees iscuss a possible implementation of

More information

Data Structure. IBPS SO (IT- Officer) Exam 2017

Data Structure. IBPS SO (IT- Officer) Exam 2017 Data Structure IBPS SO (IT- Officer) Exam 2017 Data Structure: In computer science, a data structure is a way of storing and organizing data in a computer s memory so that it can be used efficiently. Data

More information

Definitions A A tree is an abstract t data type. Topic 17. "A tree may grow a. its leaves will return to its roots." Properties of Trees

Definitions A A tree is an abstract t data type. Topic 17. A tree may grow a. its leaves will return to its roots. Properties of Trees Topic 17 Introduction ti to Trees "A tree may grow a thousand feet tall, but its leaves will return to its roots." -Chinese Proverb Definitions A A tree is an abstract t data t d internal type nodes one

More information

Chapter 20: Binary Trees

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

More information

CSI33 Data Structures

CSI33 Data Structures Outline Department of Mathematics and Computer Science Bronx Community College November 13, 2017 Outline Outline 1 C++ Supplement.1: Trees Outline C++ Supplement.1: Trees 1 C++ Supplement.1: Trees Uses

More information

CSC148H Week 8. Sadia Sharmin. July 12, /29

CSC148H Week 8. Sadia Sharmin. July 12, /29 CSC148H Week 8 Sadia Sharmin July 12, 2017 1/29 Motivating Trees I A data structure is a way of organizing data I Stacks, queues, and lists are all linear structures I They are linear in the sense that

More information

M-ary Search Tree. B-Trees. B-Trees. Solution: B-Trees. B-Tree: Example. B-Tree Properties. Maximum branching factor of M Complete tree has height =

M-ary Search Tree. B-Trees. B-Trees. Solution: B-Trees. B-Tree: Example. B-Tree Properties. Maximum branching factor of M Complete tree has height = M-ary Search Tree B-Trees Section 4.7 in Weiss Maximum branching factor of M Complete tree has height = # disk accesses for find: Runtime of find: 2 Solution: B-Trees specialized M-ary search trees Each

More information

Data Structures and Algorithms

Data Structures and Algorithms Data Structures and Algorithms Trees Sidra Malik sidra.malik@ciitlahore.edu.pk Tree? In computer science, a tree is an abstract model of a hierarchical structure A tree is a finite set of one or more nodes

More information

UNIT III TREES. A tree is a non-linear data structure that is used to represents hierarchical relationships between individual data items.

UNIT III TREES. A tree is a non-linear data structure that is used to represents hierarchical relationships between individual data items. UNIT III TREES A tree is a non-linear data structure that is used to represents hierarchical relationships between individual data items. Tree: A tree is a finite set of one or more nodes such that, there

More information

BBM 201 Data structures

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

More information

a graph is a data structure made up of nodes in graph theory the links are normally called edges

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

More information

Successor/Predecessor Rules in Binary Trees

Successor/Predecessor Rules in Binary Trees Successor/Predecessor Rules in inary Trees Thomas. nastasio July 7, 2003 Introduction inary tree traversals are commonly made in one of three patterns, inorder, preorder, and postorder. These traversals

More information

Friday Four Square! 4:15PM, Outside Gates

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.

More information

Trees, Binary Trees, and Binary Search Trees

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,

More information

Garbage Collection: recycling unused memory

Garbage Collection: recycling unused memory Outline backtracking garbage collection trees binary search trees tree traversal binary search tree algorithms: add, remove, traverse binary node class 1 Backtracking finding a path through a maze is an

More information

CS301 - Data Structures Glossary By

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

More information

Course Review for. Cpt S 223 Fall Cpt S 223. School of EECS, WSU

Course Review for. Cpt S 223 Fall Cpt S 223. School of EECS, WSU Course Review for Midterm Exam 1 Cpt S 223 Fall 2011 1 Midterm Exam 1 When: Friday (10/14) 1:10-2pm Where: in class Closed book, closed notes Comprehensive Material for preparation: Lecture slides & in-class

More information

6-TREE. Tree: Directed Tree: A directed tree is an acyclic digraph which has one node called the root node

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

More information

An undirected graph is a tree if and only of there is a unique simple path between any 2 of its vertices.

An undirected graph is a tree if and only of there is a unique simple path between any 2 of its vertices. Trees Trees form the most widely used subclasses of graphs. In CS, we make extensive use of trees. Trees are useful in organizing and relating data in databases, file systems and other applications. Formal

More information

(2,4) Trees Goodrich, Tamassia (2,4) Trees 1

(2,4) Trees Goodrich, Tamassia (2,4) Trees 1 (2,4) Trees 9 2 5 7 10 14 2004 Goodrich, Tamassia (2,4) Trees 1 Multi-Way Search Tree A multi-way search tree is an ordered tree such that Each internal node has at least two children and stores d -1 key-element

More information

CISC 235 Topic 3. General Trees, Binary Trees, Binary Search Trees

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

More information

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 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

More information

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 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

More information

COMP 250 Fall binary trees Oct. 27, 2017

COMP 250 Fall binary trees Oct. 27, 2017 The order of a (rooted) tree is the maximum number of children of any node. A tree of order n is called an n-ary tree. It is very common to use trees of order 2. These are called binary trees. Binary Trees

More information

M-ary Search Tree. B-Trees. Solution: B-Trees. B-Tree: Example. B-Tree Properties. B-Trees (4.7 in Weiss)

M-ary Search Tree. B-Trees. Solution: B-Trees. B-Tree: Example. B-Tree Properties. B-Trees (4.7 in Weiss) M-ary Search Tree B-Trees (4.7 in Weiss) Maximum branching factor of M Tree with N values has height = # disk accesses for find: Runtime of find: 1/21/2011 1 1/21/2011 2 Solution: B-Trees specialized M-ary

More information

Binary search trees (BST) Binary search trees (BST)

Binary search trees (BST) Binary search trees (BST) Tree A tree is a structure that represents a parent-child relation on a set of object. An element of a tree is called a node or vertex. The root of a tree is the unique node that does not have a parent

More information

Trees. Reading: Weiss, Chapter 4. Cpt S 223, Fall 2007 Copyright: Washington State University

Trees. Reading: Weiss, Chapter 4. Cpt S 223, Fall 2007 Copyright: Washington State University Trees Reading: Weiss, Chapter 4 1 Generic Rooted Trees 2 Terms Node, Edge Internal node Root Leaf Child Sibling Descendant Ancestor 3 Tree Representations n-ary trees Each internal node can have at most

More information

CSCI2100B Data Structures Trees

CSCI2100B Data Structures Trees CSCI2100B Data Structures Trees Irwin King king@cse.cuhk.edu.hk http://www.cse.cuhk.edu.hk/~king Department of Computer Science & Engineering The Chinese University of Hong Kong Introduction General Tree

More information

EE 368. Weeks 5 (Notes)

EE 368. Weeks 5 (Notes) EE 368 Weeks 5 (Notes) 1 Chapter 5: Trees Skip pages 273-281, Section 5.6 - If A is the root of a tree and B is the root of a subtree of that tree, then A is B s parent (or father or mother) and B is A

More information

Analysis of Algorithms

Analysis of Algorithms Algorithm An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. A computer program can be viewed as an elaborate algorithm. In mathematics and

More information

CSI33 Data Structures

CSI33 Data Structures Outline Department of Mathematics and Computer Science Bronx Community College October 19, 2016 Outline Outline 1 Chapter 7: Trees Outline Chapter 7: Trees 1 Chapter 7: Trees Uses Of Trees Chapter 7: Trees

More information

COSC 2007 Data Structures II Final Exam. Part 1: multiple choice (1 mark each, total 30 marks, circle the correct answer)

COSC 2007 Data Structures II Final Exam. Part 1: multiple choice (1 mark each, total 30 marks, circle the correct answer) COSC 2007 Data Structures II Final Exam Thursday, April 13 th, 2006 This is a closed book and closed notes exam. There are total 3 parts. Please answer the questions in the provided space and use back

More information

INF2220: algorithms and data structures Series 1

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

More information

Bioinformatics Programming. EE, NCKU Tien-Hao Chang (Darby Chang)

Bioinformatics Programming. EE, NCKU Tien-Hao Chang (Darby Chang) Bioinformatics Programming EE, NCKU Tien-Hao Chang (Darby Chang) 1 Tree 2 A Tree Structure A tree structure means that the data are organized so that items of information are related by branches 3 Definition

More information

Programming II (CS300)

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

More information

Searching in Graphs (cut points)

Searching in Graphs (cut points) 0 November, 0 Breath First Search (BFS) in Graphs In BFS algorithm we visit the verices level by level. The BFS algorithm creates a tree with root s. Once a node v is discovered by BFS algorithm we put

More information

Trees (Part 1, Theoretical) CSE 2320 Algorithms and Data Structures University of Texas at Arlington

Trees (Part 1, Theoretical) CSE 2320 Algorithms and Data Structures University of Texas at Arlington Trees (Part 1, Theoretical) CSE 2320 Algorithms and Data Structures University of Texas at Arlington 1 Trees Trees are a natural data structure for representing specific data. Family trees. Organizational

More information

Data Structures. Trees. By Dr. Mohammad Ali H. Eljinini. M.A. Eljinini, PhD

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

More information

The tree data structure. Trees COL 106. Amit Kumar Shweta Agrawal. Acknowledgement :Many slides are courtesy Douglas Harder, UWaterloo

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

More information

Trees and Tree Traversal

Trees and Tree Traversal Trees and Tree Traversal Material adapted courtesy of Prof. Dave Matuszek at UPENN Definition of a tree A tree is a node with a value and zero or more children Depending on the needs of the program, the

More information

INF2220: algorithms and data structures Series 1

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:

More information