TREE
Ahsan Ahmed
College of Computer and Information Science
Majma’ah University
1
Tree in C++
 Outlines:
What is a Tree
Tree Terminology
ADT: Tree
Representation of Tree Node
Left Child, Right Sibling Representation
Tree Traversal
 Preorder Traversal
 Postorder Traversal
 Inorder Traversal
2
Tree in C++
 Outlines:
Binary Tree
ADT: Binary Tree
Differences Between A Tree and A Binary Tree
Data Structure for Binary Trees
Arithmetic Expression Tree
Maximum Number of Nodes in a Binary Tree
Full Binary Tree
3
Tree in C++
 What is a Tree
 A tree is a finite nonempty set of elements.
 It is an abstract model of a hierarchical structure.
 consists of nodes with a parent-child relation.
 Applications:
Computers”R”Us
 Organization charts
 File systems
 Programming environments
Local
4
Europe
Sales
Manufacturing
International
Asia
Laptops
Canada
Desktops
R&D
Tree in C++
 Tree Terminology
 Root: node without parent (A)
 Siblings: nodes share the same parent
 Internal node: node with at least one child (A, B, C, F)
 External node (leaf): node without children (E, I, J, K, G, H, D)
 Ancestors of a node: parent, grandparent, grand-grandparent, etc.
 Descendant of a node: child, grandchild, grand-grandchild, etc.
A
B
E
5
D
C
F
G
H
subtree
I
J
K
Tree in C++
 Tree Terminology …. Cont.
 Depth of a node: number of ancestors
 Height of a tree: maximum depth of any node (3)
 Degree of a node: the number of its children
 Degree of a tree: the maximum number of its node.
 Subtree: tree consisting of a node and its descendants.
A
B
E
6
D
C
F
G
H
subtree
I
J
K
Tree in C++
 ADT: Tree
 We use positions (p) to abstract nodes
 Generic methods:
 integer size()
 boolean isEmpty()
 objectIterator elements()
 positionIterator positions()
 Accessor methods:
 position root()
 position parent(p)
 positionIterator children(p)
7
Tree in C++
 ADT: Tree … Cont.
 Query methods:
 boolean isInternal(p)
 boolean isExternal(p)
 boolean isRoot(p)
 Update methods:
 swapElements(p, q)
 object replaceElement(p, o)
 Additional update methods may be defined by data structures
implementing the Tree ADT
8
Tree in C++
 Representation of Tree Node
 List Representation
 ( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) )
 The root comes first, followed by a list of links to sub-trees
How many link fields are needed in
such a representation?
Data
9
Link 1
Link 2
…
Link n
Tree in C++
 Representation of Tree Node
 Every tree node:
 object – useful information
 children – pointers to its children
Data
Data

Data
10


Data


Data

Data


Data



Tree in C++
 Representation of Tree Node
 A node is represented by an object storing
 Element.

 Parent node.
 Sequence of children nodes.
B

B
C
11
A
D
A

D
F
F
E

C

E
Tree in C++
 Left Child, Right Sibling Representation
Data
Left
Child
A
Right
Sibling
B
E
J
12
F
K
C
D
G
H
L
I
Tree in C++
 Tree Traversal
 Two main methods:
 Preorder
 Postorder
 Preorder:
 visit the root
 traverse in preorder the children (subtrees)
13
 Postorder
 traverse in postorder the children (subtrees)
 visit the root
Tree in C++
 Preorder Traversal
 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
Become Rich
2
5
1. Motivations
14
9
2. Methods
3
4
1.1 Enjoy
Life
1.2 Help
Poor Friends
6
2.1 Get a
CS BSc
7
2.2 Start a
Web Site
3. Success Stories
8
2.3 Acquired
by Google
Tree in C++
 Postorder Traversal
 In a postorder traversal, a node is visited after its descendants
 Application: compute space used by files in a directory and its
subdirectories
9
cs16/
3
7
homeworks/
15
8
todo.txt
1K
programs/
1
2
h1c.doc
3K
h1nc.doc
2K
4
DDR.cpp
10K
5
Stocks.cpp
25K
6
Robot.cpp
20K
Tree in C++
 Inorder Traversal
 In an inorder traversal a node is visited after its left subtree and
before its right subtree.
6
2
8
1
4
3
16
7
5
9
Tree in C++
 Binary Tree
 A binary tree is a tree with the following properties:
 Each internal node has at most two children (degree of two).
 The children of a node are an ordered pair.
 We call the children of an internal node left child and right child.
 Alternative recursive definition: a binary tree is either
 a tree consisting of a single node, OR
 a tree whose root has an ordered pair of children, each of which is a
binary tree
A
 Applications:
 arithmetic expressions
B
C
 decision processes
D
E
F
G
 searching
17
H
I
Tree in C++
 ADT: Binary Tree
 The Binary Tree ADT extends the Tree ADT, i.e., it inherits all the
methods of the Tree ADT.
 Additional methods:
 position leftChild(p)
 position rightChild(p)
 position sibling(p)
 Update methods may be defined by data structures implementing
the BinaryTree ADT.
18
Tree in C++
 ADT: Binary Tree
 Examples of the Binary Tree
Complete Binary Tree
Skewed Binary Tree
A
B
1
A
A
2
B
B
C
C
3
D
E
D
19
E
5
4
H
I
F
G
Tree in C++
 Differences Between A Tree and A Binary Tree
a) The subtrees of a binary tree are ordered; those of a tree are not
ordered.
b) Binary tree can be empty while general tree can not.
c) Nodes in binary tree can not more than 2 where as in tree there is no
limit.
A
B
A
B
Are different when viewed as binary trees.
Are the same when viewed as trees.
20
Tree in C++
 Data Structure for Binary Trees
 A node is represented by an object storing
 Element

 Parent node
 Left child node
B
 Right child node

B
A

A
D
D

C
21



E
C
E
Tree in C++
 Arithmetic Expression Tree
 Binary tree associated with an arithmetic expression
 internal nodes: operators
 external nodes: operands
 Example: arithmetic expression tree for the expression
(2 * (a - 1) + (3 * b))
+


-
2
22
a
3
1
b
 Infix:
 All binary operators appear between the two operands
 Example: ((1+2) * (3-4))
 Prefix:
 All operators appear near the beginning of the equation, with
the two operands appearing right after.
 Example: * + 1 2 – 3 4
 Postfix:
 The operators will appear near the end
 Example: 1 2 + 3 4 - *
Assignment: (9 + (12 / 3)) * ((12 /4) – 2)
23
Tree in C++
 Decision Tree
 Binary tree associated with a decision process
 internal nodes: questions with yes/no answer
 external nodes: decisions
 Example: dining decision
Want a fast meal?
No
Yes
How about drink?
24
How about shopping?
Yes
No
Yes
Fruits
Coffee
Drive to City center
No
Continue driving
Tree in C++
 Maximum Number of Nodes in a Binary Tree
 The maximum number of nodes on depth i of a binary tree is 2i,
where i>=0.
 The maximum nubmer of nodes in a binary tree of height k is
2k+1-1, where k>=0.
Prove by induction
k
i
k +1
2

2
-1

25
i 0
Tree in C++
 Full Binary Tree
 A full binary tree of a given height k has 2k+1–1 nodes.
26
Height 3 full binary tree.