CSECE 268 DATA STRUCTURE

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CS/ECE 268 DATA STRUCTURE

• SOUNDARARAJAN EZEKIEL
• DEPARTMENT OF COMPUTERSCIENCE
• OHIO NORTHERN UNIVERSITY

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Trees

• Productivity experts says that breakthrough come
  by thinking nonlinearity
• one of the most important nonlinear data
  structure in computing -- tree
• It is a breakthrough in data organization
• algorithms are much faster than when using linear
  data structure such as list, vector, and sequences

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• Tree also provide a natural organization for data
• file system
• graphical user interface
• database
• Web sites
• other computer systems
• we say trees are non-linear, we are referring to
  an organizational relationship that is richer
  than the simple before and after
  relationship between the objects in sequences

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• The relationship in a tree are hierarchical---
  with some objects being above and some below
  others
• Actually the main terminology for tree data
  structure comes from family trees, with the terms
  parents child, ancestor, and descendant
  being the most common words used to describe
  relationship
• We show an example of a family tree from Holy
  Bible

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Example Genesis Chapter 25-36
Eldaah Abida Hanoch Epher Ephah Dedan Sheba
Shuah Ishbak Midian Medan Jokshan Zimran Ishama
el Isaac
Eliphaz Revel Jeusuh Jalam Korah
Nebaioth Kedar Adbeel Mibsam Mishma Dumah Massa Ha
dad tema Jetur Naphish Jedemah
Benjamin Joseph Dinah Zebulun Issachar Asher Gad N
aphtali Dan Judah levi Simeon Reuben
Abraham
Esau Jacob(Israel)
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The tree ADT(abstract data type)

• A tree is an ADT that stores elements
  hierarchically
• With exception of the top element, each element
  in a tree has a parent element and zero or more
  children elements
• tree is visualized by placing elements inside
  ovals or rectangles, and drawing the connections
  between parents and children with straight lines
• We typically call the top element the root of the
  tree, but is drawn as the highest element, with
  the other element being connected below

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Example
ONU RUS
Manu
Sales
purchase
RD
TV
CD
computer
Internat
Domestic
Overseas
Canada
S.Ameri
Africa
Europe
Asia
Australia
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Terminology and Basic Properties

• A tree T is a set of nodes storing elements in a
  parent-child relationship with the following
  properties
• T has a special node r, called the root of T
• Each node v of T different from r has a parent
  node u

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Caution

• Note that according to the above definition, a
  tree cannot be empty, since it must have at least
  one node, the root
• Some people allow the definition to include the
  empty tree, but we adopt the convention that a
  tree always has a root so as to keep our
  presentation simple and to avoid having to always
  deal with the special case of an empty tree in
  our algorithm

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More Terminology

• if node u is the parent of node v, then we say
  that v is a child of u
• two nodes that are children of the same parent
  are sibling
• a node is external if it has no children and it
  is internal if it has one or more children
• external nodes are also known as leaves
• the subtree of T rooted at a node v is the tree
  consisting of all the descendents of v in
  T(including v itself)

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• An ancestor of a node is either the node itself
  or an ancestor of the parent of the node
• Conversely we say that a node v is a descendent
  of a node u if u is an ancestor of v
• Example Sales is an ancestor of Europe is a
  descendent of Sales

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Example 1

• The inheritance relation between classes in a
  Java program forms tree. The class.java.lang.Objec
  t is an ancestor of all other classes

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Example 2

• In most operating systems, files are organized
  hierarchically into nested directories( also
  called folders), which are presented to the user
  in the form of a tree
• More specifically, the internal nodes of the tree
  are associated with directories and the external
  nodes are associated with regular files
• In the UNIX OS the root of the tree is
  appropriately called the root directory and is
  responded by the symbol /. It is the ancestor
  of all directories and files in a UNIX file system

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/user/rt/courses/
CS 268
CS 365
Prog
HW
Projects
grades
grades
demos
papers
pr3
hw1
pr2
pr1
hw2
hw3
perfor
pipe
data
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Terminology

• A tree is ordered if there is a linear ordering
  defined for the children of each node that is,
  we can identify children of a node as being the
  first, second, third, and so on.

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Example 3

• A structural document, such as book, is
  hierarchically organized as a tree whose internal
  nodes are chapters, sections, and subsections,
  and external nodes are paragraph, tables,
  figures, the bibliography and so on..
• We expand the tree further to show paragraph
  consists of sentences, sentences consists of
  words, and words consists of characters
• In any case, such a tree is an example of an
  ordered tree, because there is well-defined
  ordering among the children of each node

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Binary tree

• a binary tree is an ordered tree in which every
  node has at most two children
• A binary tree is proper if each node has either
  zero or two children
• Thus, in a proper binary tree, every internal
  node has exactly two children
• For each internal node in a binary tree, we label
  each child as either being a left child or a
  right child.
• These children are ordered so that a left child
  comes before the a right child

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• The subtree rooted as a left or right child of an
  internal node v is called a left subtree or right
  subtree respectively of v
• Binary trees have a number of useful applications
• we will discuss two common uses in the example
  below

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Example Investment Advise
Are you nervous?
Yes
NO
Saving account
Will you need to access most of the money within
next 5 years?
NO
Yes
Are you willing to accept risk in exchange for
higher expected return?
Money market fund
NO
Yes
Diversified portfolio with stocks, bonds and
short tern instruements
Stock portfolio
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Example Arithmetic Expression

• An arithmetic expression can be represented by a
  tree whose external nodes are associated with
  variables or constants, and whose internal nodes
  are associated with one of the operators, , -,
  x, and /
• Each node in such a tree has a value associated
  with it
• If node is external, then its value is that of
  its variable or constant

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• if a node is internal, then its value is defined
  by applying its operation to the values of its
  children
• such an arithmetic expression tree is a proper
  binary tree, since each of the operators , -, x,
  and / take exactly two operands.
• Of course, if we were to allow for unary
  operators like (-) negation, as in (-x) then we
  could have an improper binary tree

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((((31)x3)/((9-5)2))-((3x(7-4))6))
-
/

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x

x
-

3
3
-
2
1
3
9
5
7
4
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Tree methods

• The tree ADT stores elements at positions, which,
  as with positions in a list, are defined relative
  to neighboring positions.
• The positions in a tree are its nodes, and
  neighboring positions satisfy the parent-child
  relationships that define a valid tree.
• we use the terms position and node
  interchangeably for trees

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• element() Return the object at this position
  Input None, Output Object
• the real power of node positions in a tree,
  however, comes from the accessor methods of the
  tree ADT that return and accept positions, such
  as the follwoing

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• root() Return the root of the tree Input
  None, Output Position
• parent(v) Return the parent of node v, an error
  occurs if v is not root, Input Position
  Output Position
• children(v) Return an iterator of the children
  of node v. Input Position, Output Iterator of
  positions

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Iterator

• A typical computation on a vector, list, or
  sequence is to march through its elements in
  order, one at a time, for example, to look for a
  specific element
• An iterator is asoftware design pattern that
  abstracts the process of scanning though a
  collection of elements one element at a time. An
  iterator consists of a sequence S, a current
  position in S, and a way of stepping to the next
  position in S and making it the current position

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