CS210- Lecture 9
June 20, 2005
Announcements
 Assignment 3 has been posted
 Part 1 is due next Monday
 Part 2 is due next Tuesday
 Midterm is on June 30
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Section 3.1 - 3.4
Chapter 4
Section 5.1 - 5.3, Section 5.5
Part of Chapter 6 that I will be able to cover till this
Thursday (06/23).
 I will return Assignment 1 on Thursday (06/23)
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Agenda
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Assignment 3
Tree ADT
Tree Terminology
Tree Methods
Tree Interface
Tree Traversals
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Assignment 3
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A linked list does not truly provide opportunity to use
unbounded memory.
In reality, memory for a system can be viewed as one large,
consecutive array of memory cells. When programming in
Java, or most other high-level languages, the systems takes
care of most of the details of managing this memory.
For example, when you, as a programmer, want memory for
creating an additional node in a list, you might give an
instruction such as,
Node extra = new Node();
When interpreting this instruction, Java finds available cells
of memory which can be used by you for an object of class
Node.
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Singly Linked List embedded in Array
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Head
Index
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Doubly Linked List embedded in Array
Array
Index
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Array
Contents
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-23
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Header Index: 12
Header  17  -23  0  4  Trailer
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Tree ADT
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Tree is an abstract data type that stores
elements hierarchically.
Except top element, each element in a
tree has a parent element and zero or
more children elements.
The top element is called the root of
the tree.
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What is a Tree
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A tree T consists of
nodes with a parentchild relationship that
satisfies following
properties:
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If T is nonempty, it has
a special node, called
the root of T, that has
no parent.
Each node v of T
different from the root
E
has a unique parent
node w; every node
with a parent w is a
child of w.
Applications:
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A
I
B
C
F
J
G
D
H
K
Organization charts
File systems
Programming
environments
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Tree Terminology
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Root: node without parent (A)
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Subtree: tree consisting of
a node and its
Internal node: node with at least
descendants
one child (A, B, C, F)
External node (a.k.a. leaf ): node
A
without children (E, I, J, K, G, H, D)
Ancestors of a node: parent,
grandparent, grand-grandparent,
B
C
D
etc.
Depth of a node: number of
ancestors
E
F
G
H
Height of a tree: maximum depth
of any node (3)
Descendant of a node: child,
I
J
K
subtree
grandchild, grand-grandchild, etc.
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Tree Terminology
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Two nodes that are children of the
same parent are siblings.
A node u is an ancestor of a node v if u
= v or u is an ancestor of the parent of
v.
The inheritance relation between
classes in a java program forms a tree.
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Ordered Tree
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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.
A binary tree is an ordered tree with the
following properties:
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Each node has at most two children.
Each child node is labeled as being either a left
child or a right child.
The left child precedes the right child in the
ordering of the children of a node.
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Recursive def. of Binary Tree
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Edge
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A node r, called the root of T and storing an
element.
A binary tree, called the left subtree of T.
A binary tree, called the right subtree of T.
An edge of tree T is a pair of nodes (u, v) such
that u is the parent of v.
Path
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A path of T is a sequence of nodes such that any
two consecutive nodes in the sequence form an
edge.
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Applications of Binary tree
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An arithmetic expression can be represented
by a binary tree whose external nodes are
associated with variables or constants, and
whose internal nodes are associated with one
of the operators +, - , *, and /. Each node in
a tree has a value associated with it:
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If a node is external, then its value is that of its
variable or constant.
If a node is internal, then its value is defined by
applying its operation to the value of its children.
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Tree ADT
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We use positions to
abstract nodes
Generic methods:
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integer size()
boolean isEmpty()
Iterator elements()
Iterator positions()
Accessor methods:
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position root()
position parent(p)
positionIterator
children(p)
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Query methods:
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Update method:
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boolean isInternal(p)
boolean isExternal(p)
boolean isRoot(p)
object replace (p, o)
Additional update methods
may be defined by data
structures implementing the
Tree ADT
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Performance Assumptions
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The accessor methods root() and parent (v)
take O(1) time.
Query methods isInternal(v), isExternal(v)
and isRoot(v) takes O(1) time as well.
The accessor method children(v) takes O(ci)
time, where ci is the number of children of v.
The generic methods size() and isEmpty()
take O(1) time.
The generic methods replace(v, e) runs in
O(1) time.
The generic methods elements() and
positions() take O(n) time where n is the
number of nodes.
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Basic Algorithms on Trees
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Depth
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The depth of a node v can be recursively
defined as follows:
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If v is the root, then the depth of v is 0.
Otherwise, the depth of v is one plus the depth
of the parent v.
Try writing depth (Tree T, Position v)
Draw recursive trace.
Worst case running time is O(n) because in the
worst case the depth might be n – 1.
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Basic Algorithms on Trees
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Height
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The height of a node v can be recursively defined
as follows:
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If v is an external node, then the height of v is 0.
Otherwise, the height of v is one plus the maximum
height of a child of v.
The height of a nonempty tree T is equal to the
maximum depth an external node of T.
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Try writing height1 (Tree T) non recursively by calling
depth.
Worst case running time is O(n2).
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Height of a Tree
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Try writing height2 (Tree T, Position v)
using recursive definition of height of a
node.
The height of the tree is computed by
calling
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Height2(T, T.root())
This is more efficient than non recursive
method: O(n)
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Preorder Traversal
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A traversal visits the nodes of a
tree in a systematic manner
In a preorder traversal, a node is
visited before its descendants
Application: print a structured
document
1
2
Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)
A
5
9
B
E
3
4
C
D
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Preorder Traversal
public static String preorderPrint (Tree T, Position v)
{
String s = v.element().toString();
Iterator children = T.children(v);
while(children.hasNext())
S+= “ ” + preorderPrint (T, (Position) children.next());
return s;
}
Preorder Traversal runs in O(n) time.
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Postorder Traversal
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In a postorder traversal, a
node is visited after its
descendants
Application: compute space
used by files in a directory and
its subdirectories
Try writing postorderPrint.
9
3
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v)
A
8
7
B
E
1
2
C
D
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