This document provides an overview of tree data structures and binary trees. It begins by defining trees and their basic concepts such as subtrees, leaves, levels, and roots. It then defines binary trees and contrasts them with general trees. The document discusses calculating the height of full binary trees and using trees to represent arithmetic expressions. It also covers traversing trees and different ways of representing trees in memory.

1. Data Structures
Trees
Andres Mendez-Vazquez
May 6, 2015
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Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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Trees Structures
Linear lists are useful for serially ordered data.
(e0,e1,...,en−1)
Days of week.
Months in a year.
Students in this class.
Trees are useful for hierarchically ordered data
Employees of a corporation.
President, vice presidents, managers, and so on.
Java's classes.
Object is at the top of the hierarchy.
Subclasses of Object are next, and so on.
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Trees Structures
Linear lists are useful for serially ordered data.
(e0,e1,...,en−1)
Days of week.
Months in a year.
Students in this class.
Trees are useful for hierarchically ordered data
Employees of a corporation.
President, vice presidents, managers, and so on.
Java's classes.
Object is at the top of the hierarchy.
Subclasses of Object are next, and so on.
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Denition
First
A tree T is a nite nonempty set of elements.
Root
One of these elements is called the root.
The rest
The remaining elements, if any, are partitioned into trees, which are called
the subtrees of T.
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Denition
First
A tree T is a nite nonempty set of elements.
Root
One of these elements is called the root.
The rest
The remaining elements, if any, are partitioned into trees, which are called
the subtrees of T.
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Denition
First
A tree T is a nite nonempty set of elements.
Root
One of these elements is called the root.
The rest
The remaining elements, if any, are partitioned into trees, which are called
the subtrees of T.
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For example
Class Structure
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For example
Class Structure
ROOT
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For example
Class Structure
ROOT
Children of The ROOT
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For example
Class Structure
ROOT
Children of The ROOT
Grand Children of The ROOT
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Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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SubTrees
We have the following
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Leaves
We have the following
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Levels
We have the following
Level 4
Level 1
Level 2
Level 3
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Caution
BE AWARE
Some texts start level numbers at 0 rather than at 1.
Thus
Root is at level 0. Its children are at level 1.
The grand children of the root are at level 2.
And so on.
We will adapt to the following notation
Root is at level 1!!!
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Caution
BE AWARE
Some texts start level numbers at 0 rather than at 1.
Thus
Root is at level 0. Its children are at level 1.
The grand children of the root are at level 2.
And so on.
We will adapt to the following notation
Root is at level 1!!!
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Caution
BE AWARE
Some texts start level numbers at 0 rather than at 1.
Thus
Root is at level 0. Its children are at level 1.
The grand children of the root are at level 2.
And so on.
We will adapt to the following notation
Root is at level 1!!!
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19. Images/cinvestav-
Caution
BE AWARE
Some texts start level numbers at 0 rather than at 1.
Thus
Root is at level 0. Its children are at level 1.
The grand children of the root are at level 2.
And so on.
We will adapt to the following notation
Root is at level 1!!!
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20. Images/cinvestav-
Caution
BE AWARE
Some texts start level numbers at 0 rather than at 1.
Thus
Root is at level 0. Its children are at level 1.
The grand children of the root are at level 2.
And so on.
We will adapt to the following notation
Root is at level 1!!!
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General Structure
AHHHH!!!!
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Thus, we have the concept of height
Which is Height == Depth
Level 4
Level 1
Level 2
Level 3
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In addition...
Node Degree == Number Of Children
3
2 1 1
00 1 0
0
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Tree Degree == Max Node Degree
In our case 3
3
2 1 1
00 1 0
0
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Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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Denition
We have that a Binary Tree is
A nite (possibly empty) collection of elements.
Second
1 A nonempty binary tree has a root element.
2 The remaining elements (if any) are partitioned into two binary trees.
3 These are called the left and right subtrees of the binary tree.
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Denition
We have that a Binary Tree is
A nite (possibly empty) collection of elements.
Second
1 A nonempty binary tree has a root element.
2 The remaining elements (if any) are partitioned into two binary trees.
3 These are called the left and right subtrees of the binary tree.
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Denition
We have that a Binary Tree is
A nite (possibly empty) collection of elements.
Second
1 A nonempty binary tree has a root element.
2 The remaining elements (if any) are partitioned into two binary trees.
3 These are called the left and right subtrees of the binary tree.
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Denition
We have that a Binary Tree is
A nite (possibly empty) collection of elements.
Second
1 A nonempty binary tree has a root element.
2 The remaining elements (if any) are partitioned into two binary trees.
3 These are called the left and right subtrees of the binary tree.
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Example
A Way to See Them
ROOT
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Examples of Binary Trees
We have some examples
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Dierences Between A Tree A Binary Tree
First
No node in a binary tree may have a degree more than 2, whereas there is
no limit on the degree of a node in a tree.
Second
A binary tree may be empty; a tree cannot be empty.
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Dierences Between A Tree A Binary Tree
First
No node in a binary tree may have a degree more than 2, whereas there is
no limit on the degree of a node in a tree.
Second
A binary tree may be empty; a tree cannot be empty.
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Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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The height of full or complete trees.
Something Notable
FULL Tree Height
1
Number Of Nodes
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The height of full or complete trees.
Something Notable
FULL Tree Height
2
Number Of Nodes
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The height of full or complete trees.
Something Notable
FULL Tree Height
3
Number Of Nodes
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Now, if n is the number of nodes in a full tree,
Number of nodes
n = 2h
−1 (1)
Height of the tree
h = log2 (n +1) (2)
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Now, if n is the number of nodes in a full tree,
Number of nodes
n = 2h
−1 (1)
Height of the tree
h = log2 (n +1) (2)
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40. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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Arithmetic Expressions
Imagine the following expressions
(a+b)∗(c +d)+ef /g ∗h +3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c,d,e,f ,g,h,3.25,(a+b),(c +d),etc. (5)
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Arithmetic Expressions
Imagine the following expressions
(a+b)∗(c +d)+ef /g ∗h +3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c,d,e,f ,g,h,3.25,(a+b),(c +d),etc. (5)
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Arithmetic Expressions
Imagine the following expressions
(a+b)∗(c +d)+ef /g ∗h +3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c,d,e,f ,g,h,3.25,(a+b),(c +d),etc. (5)
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In addition
Delimiters
(...) (6)
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Operators Degree
Denition
Number of operands that the operator requires.
Binary operator requires two operands.
a + b
c / d
e - f
Unary operator requires one operand.
+ g
- h
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Operators Degree
Denition
Number of operands that the operator requires.
Binary operator requires two operands.
a + b
c / d
e - f
Unary operator requires one operand.
+ g
- h
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Operators Degree
Denition
Number of operands that the operator requires.
Binary operator requires two operands.
a + b
c / d
e - f
Unary operator requires one operand.
+ g
- h
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About the Inx form
Something Notable
Normal way to write an expression.
Thus
Binary operators come in between their left and right operands.
Example
a * b
a + b * c a * b / c (a + b) * (c + d) + e f/g*h + 3.25
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About the Inx form
Something Notable
Normal way to write an expression.
Thus
Binary operators come in between their left and right operands.
Example
a * b
a + b * c a * b / c (a + b) * (c + d) + e f/g*h + 3.25
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50. Images/cinvestav-
About the Inx form
Something Notable
Normal way to write an expression.
Thus
Binary operators come in between their left and right operands.
Example
a * b
a + b * c a * b / c (a + b) * (c + d) + e f/g*h + 3.25
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Operator Priorities
How do you gure out the operands of an operator?
a + b * c
a * b + c / d
This is done by assigning operator priorities
priority(*) = priority(/) priority(+) = priority(-)
Thus
When an operand lies between two operators, the operand associates with
the operator that has higher priority.
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Operator Priorities
How do you gure out the operands of an operator?
a + b * c
a * b + c / d
This is done by assigning operator priorities
priority(*) = priority(/) priority(+) = priority(-)
Thus
When an operand lies between two operators, the operand associates with
the operator that has higher priority.
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Operator Priorities
How do you gure out the operands of an operator?
a + b * c
a * b + c / d
This is done by assigning operator priorities
priority(*) = priority(/) priority(+) = priority(-)
Thus
When an operand lies between two operators, the operand associates with
the operator that has higher priority.
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Tie Breakers
However this can happen
When an operand lies between two operators that have the same priority,
the operand associates with the operator on the left.
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Delimiters
What about ()
Subexpression within delimiters is treated as a single operand, independent
from the remainder of the expression.
Example
(a + b) * (c d) / (e f)
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Delimiters
What about ()
Subexpression within delimiters is treated as a single operand, independent
from the remainder of the expression.
Example
(a + b) * (c d) / (e f)
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HOWEVER
Inx Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more dicult than is necessary.
Postx and prex expression forms do not rely on operator priorities, a
tie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in these
forms.
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HOWEVER
Inx Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more dicult than is necessary.
Postx and prex expression forms do not rely on operator priorities, a
tie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in these
forms.
36 / 100

59. Images/cinvestav-
HOWEVER
Inx Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more dicult than is necessary.
Postx and prex expression forms do not rely on operator priorities, a
tie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in these
forms.
36 / 100

60. Images/cinvestav-
HOWEVER
Inx Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more dicult than is necessary.
Postx and prex expression forms do not rely on operator priorities, a
tie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in these
forms.
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For example
4*(3+17)
*
4 +
3 17
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Postx Form
We have that
The postx form of a variable or constant is the same as its inx form.
a, b, 3.25
Relative order
The relative order of operands is the same in inx and postx forms.
Thus
Operators come immediately after the postx form of their operands.
Inx = a + b
Postx = ab+
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Postx Form
We have that
The postx form of a variable or constant is the same as its inx form.
a, b, 3.25
Relative order
The relative order of operands is the same in inx and postx forms.
Thus
Operators come immediately after the postx form of their operands.
Inx = a + b
Postx = ab+
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Postx Form
We have that
The postx form of a variable or constant is the same as its inx form.
a, b, 3.25
Relative order
The relative order of operands is the same in inx and postx forms.
Thus
Operators come immediately after the postx form of their operands.
Inx = a + b
Postx = ab+
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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Example
1st One
Inx = a+b*c
Postx= abc*+
2nd One
Inx = a*b+c
Postx = ab*c+
3rd One
Inx = (a + b) * (c d) / (e + f)
Postx = (ab)+cd-*ef+/
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What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a = a @
+ a + b = a @ b +
- a = a ?
- a-b = a ? b -
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What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a = a @
+ a + b = a @ b +
- a = a ?
- a-b = a ? b -
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73. Images/cinvestav-
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a = a @
+ a + b = a @ b +
- a = a ?
- a-b = a ? b -
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What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a = a @
+ a + b = a @ b +
- a = a ?
- a-b = a ? b -
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75. Images/cinvestav-
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a = a @
+ a + b = a @ b +
- a = a ?
- a-b = a ? b -
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Postx Evaluation
How do we do the evaluation?
Scan postx expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as this
operator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in postx, operators come immediately after their
operands.
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Postx Evaluation
How do we do the evaluation?
Scan postx expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as this
operator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in postx, operators come immediately after their
operands.
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Postx Evaluation
How do we do the evaluation?
Scan postx expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as this
operator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in postx, operators come immediately after their
operands.
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Postx Evaluation
How do we do the evaluation?
Scan postx expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as this
operator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in postx, operators come immediately after their
operands.
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80. Images/cinvestav-
Postx Evaluation
How do we do the evaluation?
Scan postx expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as this
operator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in postx, operators come immediately after their
operands.
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Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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84. Images/cinvestav-
Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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85. Images/cinvestav-
Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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Example: (a + b) * (c d) / (e + f)
a b + c d - * e f + /
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Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
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Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
49 / 100

90. Images/cinvestav-
Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
49 / 100

91. Images/cinvestav-
Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
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92. Images/cinvestav-
Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
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93. Images/cinvestav-
Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
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Prex Form
Here
The prex form of a variable or constant is the same as its inx form.
a, b, 3.25
Thus
The relative order of operands is the same in inx and prex forms.
Then
Operators come immediately before the prex form of their operands.
Inx = a + b
Postx = ab+
Prex = +ab
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Then, we can use trees to represent operations
a+b
+
a b
-a
-
a
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Then, we can use trees to represent operations
a+b
+
a b
-a
-
a
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Example
(a + b) * (c d) / (e + f)
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Merits Of Binary Tree Form
We love them
The left and right operands are easy to visualize.
There are code optimization algorithms that work well with the binary
tree form of an expression.
We can use simple recursive evaluations on each expression to get the
results.
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99. Images/cinvestav-
Merits Of Binary Tree Form
We love them
The left and right operands are easy to visualize.
There are code optimization algorithms that work well with the binary
tree form of an expression.
We can use simple recursive evaluations on each expression to get the
results.
52 / 100

100. Images/cinvestav-
Merits Of Binary Tree Form
We love them
The left and right operands are easy to visualize.
There are code optimization algorithms that work well with the binary
tree form of an expression.
We can use simple recursive evaluations on each expression to get the
results.
52 / 100

101. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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Minimum Number Of Nodes
Question
Which is the minimum number of nodes in a binary tree whose height is h?
First
You have at least one node at each of the h levels.
Then
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103. Images/cinvestav-
Minimum Number Of Nodes
Question
Which is the minimum number of nodes in a binary tree whose height is h?
First
You have at least one node at each of the h levels.
Then
54 / 100

104. Images/cinvestav-
Minimum Number Of Nodes
Question
Which is the minimum number of nodes in a binary tree whose height is h?
First
You have at least one node at each of the h levels.
Then
54 / 100

105. Images/cinvestav-
Maximum Number Of Nodes
Here, all possible nodes at rst h levels are present.
Here, we have that
1+2+4+8+...+2h−1
=
2h −1
2−1
= 2h
−1 (7)
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106. Images/cinvestav-
Maximum Number Of Nodes
Here, all possible nodes at rst h levels are present.
Here, we have that
1+2+4+8+...+2h−1
=
2h −1
2−1
= 2h
−1 (7)
55 / 100

107. Images/cinvestav-
Number Of Nodes Height
We have that
Let n be the number of nodes in a binary tree whose height is h.
Then
h ≤ n ≤ 2h
−1 (8)
Thus
log2 (n +1) ≤ h ⇒ log2 (n +1) ≤ h ≤ n (9)
56 / 100

108. Images/cinvestav-
Number Of Nodes Height
We have that
Let n be the number of nodes in a binary tree whose height is h.
Then
h ≤ n ≤ 2h
−1 (8)
Thus
log2 (n +1) ≤ h ⇒ log2 (n +1) ≤ h ≤ n (9)
56 / 100

109. Images/cinvestav-
Number Of Nodes Height
We have that
Let n be the number of nodes in a binary tree whose height is h.
Then
h ≤ n ≤ 2h
−1 (8)
Thus
log2 (n +1) ≤ h ⇒ log2 (n +1) ≤ h ≤ n (9)
56 / 100

110. Images/cinvestav-
Full Binary Tree
A full binary tree of a given height h has 2
h1 nodes.
57 / 100

111. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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112. Images/cinvestav-
Numbering Nodes In A Full Binary Tree
It is possible to do the following
1 Number the nodes 1 through 2h 1.
2 Number by levels from top to bottom.
3 Within a level number from left to right.
We get the following
59 / 100

113. Images/cinvestav-
Numbering Nodes In A Full Binary Tree
It is possible to do the following
1 Number the nodes 1 through 2h 1.
2 Number by levels from top to bottom.
3 Within a level number from left to right.
We get the following
59 / 100

114. Images/cinvestav-
Numbering Nodes In A Full Binary Tree
It is possible to do the following
1 Number the nodes 1 through 2h 1.
2 Number by levels from top to bottom.
3 Within a level number from left to right.
We get the following
59 / 100

115. Images/cinvestav-
Numbering Nodes In A Full Binary Tree
It is possible to do the following
1 Number the nodes 1 through 2h 1.
2 Number by levels from top to bottom.
3 Within a level number from left to right.
We get the following
59 / 100

116. Images/cinvestav-
Node Number Properties
We have the following
The parent of node i is node i/2 , unless i = 1.
Node 1 is the root and has no parent.
What about the children
Left child of node i is node 2i, unless 2i n, where n is the number of
nodes.
If 2i n, node i has no left child.
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117. Images/cinvestav-
Node Number Properties
We have the following
The parent of node i is node i/2 , unless i = 1.
Node 1 is the root and has no parent.
What about the children
Left child of node i is node 2i, unless 2i n, where n is the number of
nodes.
If 2i n, node i has no left child.
60 / 100

118. Images/cinvestav-
Node Number Properties
We have the following
The parent of node i is node i/2 , unless i = 1.
Node 1 is the root and has no parent.
What about the children
Left child of node i is node 2i, unless 2i n, where n is the number of
nodes.
If 2i n, node i has no left child.
60 / 100

119. Images/cinvestav-
Node Number Properties
We have the following
The parent of node i is node i/2 , unless i = 1.
Node 1 is the root and has no parent.
What about the children
Left child of node i is node 2i, unless 2i n, where n is the number of
nodes.
If 2i n, node i has no left child.
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120. Images/cinvestav-
After all
We get the following
61 / 100

121. Images/cinvestav-
What about the right Children?
We have the following
Right child of node i is node 2i +1, unless 2i +1 n, where n is the
number of nodes.
If 2i +1 n, node i has no right child.
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122. Images/cinvestav-
What about the right Children?
We have the following
Right child of node i is node 2i +1, unless 2i +1 n, where n is the
number of nodes.
If 2i +1 n, node i has no right child.
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123. Images/cinvestav-
Complete Binary Tree With n Nodes
Dention
1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree dened by the nodes numbered 1 through n is the
unique n node complete binary tree.
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124. Images/cinvestav-
Complete Binary Tree With n Nodes
Dention
1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree dened by the nodes numbered 1 through n is the
unique n node complete binary tree.
63 / 100

125. Images/cinvestav-
Complete Binary Tree With n Nodes
Dention
1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree dened by the nodes numbered 1 through n is the
unique n node complete binary tree.
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126. Images/cinvestav-
Example
Complete binary tree with 10 nodes.
64 / 100

127. Images/cinvestav-
How we represent them?
We have two ways
Array Representation
Linked Representation
65 / 100

128. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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129. Images/cinvestav-
Array Representation
We do the following
Number the nodes using the numbering scheme for a full binary tree. The
node that is numbered i is stored in tree[i].
Then
67 / 100

130. Images/cinvestav-
Array Representation
We do the following
Number the nodes using the numbering scheme for a full binary tree. The
node that is numbered i is stored in tree[i].
Then
67 / 100

131. Images/cinvestav-
What if we have something like this
Right-Skewed Binary Tree
What a waste of memory!!!
68 / 100

132. Images/cinvestav-
What if we have something like this
Right-Skewed Binary Tree
What a waste of memory!!!
68 / 100

133. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
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134. Images/cinvestav-
What can we do?
Linked Representation
Each binary tree node is represented as an object whose data type is
BinaryTreeNode.
The space required by an n node binary tree is n * (space required by
one node).
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135. Images/cinvestav-
What can we do?
Linked Representation
Each binary tree node is represented as an object whose data type is
BinaryTreeNode.
The space required by an n node binary tree is n * (space required by
one node).
70 / 100

136. Images/cinvestav-
Code for a Node
We have then with Generic Key and Item
p a c k a g e T r e e ;
p u b l i c c l a s s B i n a r y T r e e N o d e
{
Key k e y ;
I t e m e l e m e n t ;
B i n a r y T r e e N o d e l e f t C h i l d ; // l e f t subtree
B i n a r y T r e e N o d e r i g h t C h i l d ; // r i g h t subtree
// c o n s t r u c t o r s and any other methods
// come here
}
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137. Images/cinvestav-
Example
Look at this
72 / 100

138. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

139. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

140. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

141. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

142. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

143. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

144. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

145. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

146. Images/cinvestav-
Which Operations
We might want to have
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Maybe, we want
Display the binary tree.
Evaluate the arithmetic expression represented by a binary tree.
Finally, we want
Obtain the inx form of an expression.
Obtain the prex form of an expression.
Obtain the postx form of an expression.
73 / 100

147. Images/cinvestav-
Outline
1 Trees: Basic Concepts
Subtrees, Leaves and Levels
2 Binary Trees
Basic Concepts
The Height Of a Binary Tree
Arithmetic Expressions
Properties
Numbering Nodes in A Full Binary Tree
Array Representation
Linked Representation
Traversals
74 / 100

148. Images/cinvestav-
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of the
binary tree.
Actually
In a traversal, each element of the binary tree is visited exactly once.
Actions
During the visit of an element, all action (make a clone, display, evaluate
the operator, etc.) with respect to this element is taken.
75 / 100

149. Images/cinvestav-
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of the
binary tree.
Actually
In a traversal, each element of the binary tree is visited exactly once.
Actions
During the visit of an element, all action (make a clone, display, evaluate
the operator, etc.) with respect to this element is taken.
75 / 100

150. Images/cinvestav-
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of the
binary tree.
Actually
In a traversal, each element of the binary tree is visited exactly once.
Actions
During the visit of an element, all action (make a clone, display, evaluate
the operator, etc.) with respect to this element is taken.
75 / 100

151. Images/cinvestav-
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100

152. Images/cinvestav-
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100

153. Images/cinvestav-
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100

154. Images/cinvestav-
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100

155. Images/cinvestav-
Remarks
First
In a traversal of a binary tree, each element of the binary tree is visited
exactly once.
Second
During the visit of an element, all action (make a clone, display, evaluate
the operator, etc.) with respect to this element is taken.
77 / 100

156. Images/cinvestav-
Remarks
First
In a traversal of a binary tree, each element of the binary tree is visited
exactly once.
Second
During the visit of an element, all action (make a clone, display, evaluate
the operator, etc.) with respect to this element is taken.
77 / 100

157. Images/cinvestav-
Preorder Traversals
Code
p u b l i c s t a t i c v o i d p r e O r d e r ( B i n a r y T r e e N o d e t )
{
i f ( t != n u l l )
{
v i s i t ( t ) ;
p r e O r d e r ( t . l e f t C h i l d ) ;
p r e O r d e r ( t . r i g h t C h i l d ) ;
}
}
78 / 100

158. Images/cinvestav-
Example I : (visit = print)
Something Notable
79 / 100

159. Images/cinvestav-
Example II : (visit = print)
Something Notable
80 / 100

160. Images/cinvestav-
Actually the preorder can be used to express the prex
form!!!
Example
81 / 100

161. Images/cinvestav-
Inorder Traversals
Code
p u b l i c s t a t i c v o i d i n O r d e r ( B i n a r y T r e e N o d e t )
{
i f ( t != n u l l )
{
i n O r d e r ( t . l e f t C h i l d ) ;
v i s i t ( t ) ;
i n O r d e r ( t . r i g h t C h i l d ) ;
}
}
82 / 100

162. Images/cinvestav-
Example I : (visit = print)
Something Notable
83 / 100

163. Images/cinvestav-
Example II : (visit = print)
Something Notable
84 / 100

164. Images/cinvestav-
Example III : (visit = print)
Inorder By Projection (Squishing)
85 / 100

165. Images/cinvestav-
Inorder Of Expression Tree
Example
Not only you get the inx form
But, we have a implicit representation of the parenthesis
86 / 100

166. Images/cinvestav-
Inorder Of Expression Tree
Example
Not only you get the inx form
But, we have a implicit representation of the parenthesis
86 / 100

167. Images/cinvestav-
Postorder Traversal
Code
p u b l i c s t a t i c v o i d p o s t O r d e r ( B i n a r y T r e e N o d e t )
{
i f ( t != n u l l )
{
p o s t O r d e r ( t . l e f t C h i l d ) ;
p o s t O r d e r ( t . r i g h t C h i l d ) ;
v i s i t ( t ) ;
}
}
87 / 100

168. Images/cinvestav-
Example I : (visit = print)
Something Notable
88 / 100

169. Images/cinvestav-
Example II : (visit = print)
Something Notable
89 / 100

170. Images/cinvestav-
Actually the preorder can be used to express the postx
form!!!
Example
90 / 100

171. Images/cinvestav-
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100

172. Images/cinvestav-
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100

173. Images/cinvestav-
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100

174. Images/cinvestav-
Now
What if we want to traverse each node in every level rst - level order
92 / 100

175. Images/cinvestav-
Level Order
Code
Which one?
The code
93 / 100

176. Images/cinvestav-
Level Order
Code
Which one?
The code
L e t t be t h e t r e e r o o t .
w h i l e ( t != n u l l )
{
v i s i t t and p u t i t s c h i l d r e n on a FIFO queue ;
remove a node from t h e FIFO queue and c a l l i t t ;
// remove r e t u r n s n u l l when queue i s empty
}
93 / 100

177. Images/cinvestav-
Binary Tree Construction
Question
Can you construct the binary tree, given two traversal sequences?
However
It depends on which two sequences are given.
Properties
94 / 100

178. Images/cinvestav-
Binary Tree Construction
Question
Can you construct the binary tree, given two traversal sequences?
However
It depends on which two sequences are given.
Properties
94 / 100

179. Images/cinvestav-
Binary Tree Construction
Question
Can you construct the binary tree, given two traversal sequences?
However
It depends on which two sequences are given.
Properties
94 / 100

180. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

181. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

182. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

183. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

184. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

185. Images/cinvestav-
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely dene a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100

186. Images/cinvestav-
What if we have Inorder And Preorder
Example
inorder = g d h b e i a f j c
preorder = a b d g h e i c f j
Process
Scan the preorder left to right using the inorder to separate left and right
subtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the right
subtree.
96 / 100

187. Images/cinvestav-
What if we have Inorder And Preorder
Example
inorder = g d h b e i a f j c
preorder = a b d g h e i c f j
Process
Scan the preorder left to right using the inorder to separate left and right
subtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the right
subtree.
96 / 100

188. Images/cinvestav-
What if we have Inorder And Preorder
Example
inorder = g d h b e i a f j c
preorder = a b d g h e i c f j
Process
Scan the preorder left to right using the inorder to separate left and right
subtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the right
subtree.
96 / 100

189. Images/cinvestav-
Then
First
preorder = a b d g h e i c f j
97 / 100

190. Images/cinvestav-
Then
First
preorder = a b d g h e i c f j
97 / 100

191. Images/cinvestav-
Cont...
d is the next root; g is in the left subtree; h is in the right subtree.
98 / 100

192. Images/cinvestav-
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
postorder = g h d i e b j f c a
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
99 / 100

193. Images/cinvestav-
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
postorder = g h d i e b j f c a
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
99 / 100

194. Images/cinvestav-
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
postorder = g h d i e b j f c a
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
99 / 100

195. Images/cinvestav-
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
postorder = g h d i e b j f c a
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
99 / 100

196. Images/cinvestav-
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
level order = a b c d e f g h i j
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
100 / 100

197. Images/cinvestav-
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
level order = a b c d e f g h i j
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
100 / 100

198. Images/cinvestav-
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
level order = a b c d e f g h i j
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
100 / 100

199. Images/cinvestav-
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and right
subtrees.
Example
inorder = g d h b e i a f j c
level order = a b c d e f g h i j
Thus
Tree root is a; gdhbei are in left subtree; fjc are in right subtree.
100 / 100