preparation data structures 10 trees
TRANSCRIPT
Data StructuresTrees
Andres Mendez-Vazquez
May 6, 2015
1 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
2 / 100
Images/cinvestav-1.jpg
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.
I President, vice presidents, managers, and so on.
Java's classes.
I Object is at the top of the hierarchy.I Subclasses of Object are next, and so on.
3 / 100
Images/cinvestav-1.jpg
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.
I President, vice presidents, managers, and so on.
Java's classes.
I Object is at the top of the hierarchy.I Subclasses of Object are next, and so on.
3 / 100
Images/cinvestav-1.jpg
De�nition
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 calledthe subtrees of T .
4 / 100
Images/cinvestav-1.jpg
De�nition
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 calledthe subtrees of T .
4 / 100
Images/cinvestav-1.jpg
De�nition
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 calledthe subtrees of T .
4 / 100
Images/cinvestav-1.jpg
For example
Class Structure
5 / 100
Images/cinvestav-1.jpg
For example
Class Structure
ROOT
6 / 100
Images/cinvestav-1.jpg
For example
Class Structure
ROOT
Children of The ROOT
7 / 100
Images/cinvestav-1.jpg
For example
Class Structure
ROOT
Children of The ROOT
Grand Children of The ROOT
8 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
9 / 100
Images/cinvestav-1.jpg
SubTrees
We have the following
10 / 100
Images/cinvestav-1.jpg
Leaves
We have the following
11 / 100
Images/cinvestav-1.jpg
Levels
We have the following
Level 4
Level 1
Level 2
Level 3
12 / 100
Images/cinvestav-1.jpg
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!!!
13 / 100
Images/cinvestav-1.jpg
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!!!
13 / 100
Images/cinvestav-1.jpg
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!!!
13 / 100
Images/cinvestav-1.jpg
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!!!
13 / 100
Images/cinvestav-1.jpg
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!!!
13 / 100
Images/cinvestav-1.jpg
General Structure
AHHHH!!!!
14 / 100
Images/cinvestav-1.jpg
Thus, we have the concept of height
Which is Height == Depth
Level 4
Level 1
Level 2
Level 3
15 / 100
Images/cinvestav-1.jpg
In addition...
Node Degree == Number Of Children
3
2 1 1
00 1 0
0
16 / 100
Images/cinvestav-1.jpg
Tree Degree == Max Node Degree
In our case 3
3
2 1 1
00 1 0
0
17 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
18 / 100
Images/cinvestav-1.jpg
De�nition
We have that a Binary Tree is
A �nite (possibly empty) collection of elements.
Second1 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.
19 / 100
Images/cinvestav-1.jpg
De�nition
We have that a Binary Tree is
A �nite (possibly empty) collection of elements.
Second1 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.
19 / 100
Images/cinvestav-1.jpg
De�nition
We have that a Binary Tree is
A �nite (possibly empty) collection of elements.
Second1 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.
19 / 100
Images/cinvestav-1.jpg
De�nition
We have that a Binary Tree is
A �nite (possibly empty) collection of elements.
Second1 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.
19 / 100
Images/cinvestav-1.jpg
Example
A Way to See Them
ROOT
20 / 100
Images/cinvestav-1.jpg
Examples of Binary Trees
We have some examples
21 / 100
Images/cinvestav-1.jpg
Di�erences Between A Tree & A Binary Tree
First
No node in a binary tree may have a degree more than 2, whereas there isno limit on the degree of a node in a tree.
Second
A binary tree may be empty; a tree cannot be empty.
22 / 100
Images/cinvestav-1.jpg
Di�erences Between A Tree & A Binary Tree
First
No node in a binary tree may have a degree more than 2, whereas there isno limit on the degree of a node in a tree.
Second
A binary tree may be empty; a tree cannot be empty.
22 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
23 / 100
Images/cinvestav-1.jpg
The height of full or complete trees.
Something Notable
FULL Tree Height
1
Number Of Nodes
24 / 100
Images/cinvestav-1.jpg
The height of full or complete trees.
Something Notable
FULL Tree Height
2
Number Of Nodes
25 / 100
Images/cinvestav-1.jpg
The height of full or complete trees.
Something Notable
FULL Tree Height
3
Number Of Nodes
26 / 100
Images/cinvestav-1.jpg
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)
27 / 100
Images/cinvestav-1.jpg
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)
27 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
28 / 100
Images/cinvestav-1.jpg
Arithmetic Expressions
Imagine the following expressions
(a+b)∗ (c+d)+ e�f /g ∗h+3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c ,d ,e, f ,g ,h,3.25,(a+b),(c+d),etc . (5)
29 / 100
Images/cinvestav-1.jpg
Arithmetic Expressions
Imagine the following expressions
(a+b)∗ (c+d)+ e�f /g ∗h+3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c ,d ,e, f ,g ,h,3.25,(a+b),(c+d),etc . (5)
29 / 100
Images/cinvestav-1.jpg
Arithmetic Expressions
Imagine the following expressions
(a+b)∗ (c+d)+ e�f /g ∗h+3.25 (3)
Operators
+,−,/,∗ (4)
Operands
a,b,c ,d ,e, f ,g ,h,3.25,(a+b),(c+d),etc . (5)
29 / 100
Images/cinvestav-1.jpg
In addition
Delimiters
(...) (6)
30 / 100
Images/cinvestav-1.jpg
Operators Degree
De�nition
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
31 / 100
Images/cinvestav-1.jpg
Operators Degree
De�nition
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
31 / 100
Images/cinvestav-1.jpg
Operators Degree
De�nition
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
31 / 100
Images/cinvestav-1.jpg
About the In�x 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
32 / 100
Images/cinvestav-1.jpg
About the In�x 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
32 / 100
Images/cinvestav-1.jpg
About the In�x 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
32 / 100
Images/cinvestav-1.jpg
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 withthe operator that has higher priority.
33 / 100
Images/cinvestav-1.jpg
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 withthe operator that has higher priority.
33 / 100
Images/cinvestav-1.jpg
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 withthe operator that has higher priority.
33 / 100
Images/cinvestav-1.jpg
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.
34 / 100
Images/cinvestav-1.jpg
Delimiters
What about �()�
Subexpression within delimiters is treated as a single operand, independentfrom the remainder of the expression.
Example
(a + b) * (c � d) / (e � f)
35 / 100
Images/cinvestav-1.jpg
Delimiters
What about �()�
Subexpression within delimiters is treated as a single operand, independentfrom the remainder of the expression.
Example
(a + b) * (c � d) / (e � f)
35 / 100
Images/cinvestav-1.jpg
HOWEVER
In�x Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more di�cult than is necessary.
Post�x and pre�x expression forms do not rely on operator priorities, atie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in theseforms.
36 / 100
Images/cinvestav-1.jpg
HOWEVER
In�x Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more di�cult than is necessary.
Post�x and pre�x expression forms do not rely on operator priorities, atie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in theseforms.
36 / 100
Images/cinvestav-1.jpg
HOWEVER
In�x Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more di�cult than is necessary.
Post�x and pre�x expression forms do not rely on operator priorities, atie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in theseforms.
36 / 100
Images/cinvestav-1.jpg
HOWEVER
In�x Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more di�cult than is necessary.
Post�x and pre�x expression forms do not rely on operator priorities, atie breaker, or delimiters.
So it is easier for a computer to evaluate expressions that are in theseforms.
36 / 100
Images/cinvestav-1.jpg
For example
4*(3+17)
*
4 +
3 17
37 / 100
Images/cinvestav-1.jpg
Post�x Form
We have that
The post�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Relative order
The relative order of operands is the same in in�x and post�x forms.
Thus
Operators come immediately after the post�x form of their operands.
In�x = a + b
Post�x = ab+
38 / 100
Images/cinvestav-1.jpg
Post�x Form
We have that
The post�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Relative order
The relative order of operands is the same in in�x and post�x forms.
Thus
Operators come immediately after the post�x form of their operands.
In�x = a + b
Post�x = ab+
38 / 100
Images/cinvestav-1.jpg
Post�x Form
We have that
The post�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Relative order
The relative order of operands is the same in in�x and post�x forms.
Thus
Operators come immediately after the post�x form of their operands.
In�x = a + b
Post�x = ab+
38 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
Example
1st One
In�x = a+b*c
I Post�x= abc*+
2nd One
In�x = a*b+c
I Post�x = ab*c+
3rd One
In�x = (a + b) * (c � d) / (e + f)
I Post�x = (ab)+cd-*ef+/
39 / 100
Images/cinvestav-1.jpg
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a => a @
I + a + b => a @ b +
- a => a ?
I - a-b => a ? b -
40 / 100
Images/cinvestav-1.jpg
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a => a @
I + a + b => a @ b +
- a => a ?
I - a-b => a ? b -
40 / 100
Images/cinvestav-1.jpg
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a => a @
I + a + b => a @ b +
- a => a ?
I - a-b => a ? b -
40 / 100
Images/cinvestav-1.jpg
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a => a @
I + a + b => a @ b +
- a => a ?
I - a-b => a ? b -
40 / 100
Images/cinvestav-1.jpg
What about Unary Operators?
After all they need to be evaluated
Everything need to be interpreted!!!
Replace with new symbols
+ a => a @
I + a + b => a @ b +
- a => a ?
I - a-b => a ? b -
40 / 100
Images/cinvestav-1.jpg
Post�x Evaluation
How do we do the evaluation?
Scan post�x expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as thisoperator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in post�x, operators come immediately after theiroperands.
41 / 100
Images/cinvestav-1.jpg
Post�x Evaluation
How do we do the evaluation?
Scan post�x expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as thisoperator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in post�x, operators come immediately after theiroperands.
41 / 100
Images/cinvestav-1.jpg
Post�x Evaluation
How do we do the evaluation?
Scan post�x expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as thisoperator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in post�x, operators come immediately after theiroperands.
41 / 100
Images/cinvestav-1.jpg
Post�x Evaluation
How do we do the evaluation?
Scan post�x expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as thisoperator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in post�x, operators come immediately after theiroperands.
41 / 100
Images/cinvestav-1.jpg
Post�x Evaluation
How do we do the evaluation?
Scan post�x expression from left to right pushing operands on to a stack.
Then
When an operator is encountered, pop as many operands as thisoperator needs.
Then, evaluate the operator.
Then, push the result on to the stack.
IMPORTANT
This works because, in post�x, operators come immediately after theiroperands.
41 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
42 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
43 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
44 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
45 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
46 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
47 / 100
Images/cinvestav-1.jpg
Example: (a + b) * (c � d) / (e + f)
a b + c d - * e f + /
48 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Pre�x Form
Here
The pre�x form of a variable or constant is the same as its in�x form.
a, b, 3.25
Thus
The relative order of operands is the same in in�x and pre�x forms.
Then
Operators come immediately before the pre�x form of their operands.
In�x = a + b
Post�x = ab+
Pre�x = +ab
49 / 100
Images/cinvestav-1.jpg
Then, we can use trees to represent operations
a+b
+
a b
-a
-
a
50 / 100
Images/cinvestav-1.jpg
Then, we can use trees to represent operations
a+b
+
a b
-a
-
a
50 / 100
Images/cinvestav-1.jpg
Example
(a + b) * (c � d) / (e + f)
51 / 100
Images/cinvestav-1.jpg
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 binarytree form of an expression.
We can use simple recursive evaluations on each expression to get theresults.
52 / 100
Images/cinvestav-1.jpg
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 binarytree form of an expression.
We can use simple recursive evaluations on each expression to get theresults.
52 / 100
Images/cinvestav-1.jpg
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 binarytree form of an expression.
We can use simple recursive evaluations on each expression to get theresults.
52 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
53 / 100
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
Full Binary Tree
A full binary tree of a given height h has 2h�1 nodes.
57 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
58 / 100
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
Node Number Properties
We have the following
The parent of node i is nodebi/2c, 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 ofnodes.
If 2i > n, node i has no left child.
60 / 100
Images/cinvestav-1.jpg
Node Number Properties
We have the following
The parent of node i is nodebi/2c, 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 ofnodes.
If 2i > n, node i has no left child.
60 / 100
Images/cinvestav-1.jpg
Node Number Properties
We have the following
The parent of node i is nodebi/2c, 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 ofnodes.
If 2i > n, node i has no left child.
60 / 100
Images/cinvestav-1.jpg
Node Number Properties
We have the following
The parent of node i is nodebi/2c, 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 ofnodes.
If 2i > n, node i has no left child.
60 / 100
Images/cinvestav-1.jpg
After all
We get the following
61 / 100
Images/cinvestav-1.jpg
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 thenumber of nodes.
If 2i +1> n, node i has no right child.
62 / 100
Images/cinvestav-1.jpg
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 thenumber of nodes.
If 2i +1> n, node i has no right child.
62 / 100
Images/cinvestav-1.jpg
Complete Binary Tree With n Nodes
De�ntion1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree de�ned by the nodes numbered 1 through n is theunique n node complete binary tree.
63 / 100
Images/cinvestav-1.jpg
Complete Binary Tree With n Nodes
De�ntion1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree de�ned by the nodes numbered 1 through n is theunique n node complete binary tree.
63 / 100
Images/cinvestav-1.jpg
Complete Binary Tree With n Nodes
De�ntion1 It starts with a full binary tree that has at least n nodes.
2 Number the nodes as described earlier.
3 The binary tree de�ned by the nodes numbered 1 through n is theunique n node complete binary tree.
63 / 100
Images/cinvestav-1.jpg
Example
Complete binary tree with 10 nodes.
64 / 100
Images/cinvestav-1.jpg
How we represent them?
We have two ways
Array Representation
Linked Representation
65 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
66 / 100
Images/cinvestav-1.jpg
Array Representation
We do the following
Number the nodes using the numbering scheme for a full binary tree. Thenode that is numbered i is stored in tree[i ].
Then
67 / 100
Images/cinvestav-1.jpg
Array Representation
We do the following
Number the nodes using the numbering scheme for a full binary tree. Thenode that is numbered i is stored in tree[i ].
Then
67 / 100
Images/cinvestav-1.jpg
What if we have something like this
Right-Skewed Binary Tree
What a waste of memory!!!
68 / 100
Images/cinvestav-1.jpg
What if we have something like this
Right-Skewed Binary Tree
What a waste of memory!!!
68 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
69 / 100
Images/cinvestav-1.jpg
What can we do?
Linked Representation
Each binary tree node is represented as an object whose data type isBinaryTreeNode.
The space required by an n node binary tree is n * (space required byone node).
70 / 100
Images/cinvestav-1.jpg
What can we do?
Linked Representation
Each binary tree node is represented as an object whose data type isBinaryTreeNode.
The space required by an n node binary tree is n * (space required byone node).
70 / 100
Images/cinvestav-1.jpg
Code for a Node
We have then with Generic Key and Item
package Tree ;p u b l i c c l a s s BinaryTreeNode{
Key key ;I tem e lement ;BinaryTreeNode l e f t C h i l d ; // l e f t s u b t r e eBinaryTreeNode r i g h t C h i l d ; // r i g h t s u b t r e e// c o n s t r u c t o r s and any o th e r methods// come he r e
}
71 / 100
Images/cinvestav-1.jpg
Example
Look at this
72 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
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 in�x form of an expression.
Obtain the pre�x form of an expression.
Obtain the post�x form of an expression.
73 / 100
Images/cinvestav-1.jpg
Outline
1 Trees: Basic ConceptsSubtrees, Leaves and Levels
2 Binary TreesBasic ConceptsThe Height Of a Binary TreeArithmetic ExpressionsPropertiesNumbering Nodes in A Full Binary TreeArray RepresentationLinked RepresentationTraversals
74 / 100
Images/cinvestav-1.jpg
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of thebinary 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, evaluatethe operator, etc.) with respect to this element is taken.
75 / 100
Images/cinvestav-1.jpg
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of thebinary 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, evaluatethe operator, etc.) with respect to this element is taken.
75 / 100
Images/cinvestav-1.jpg
Traversals
Something Notable
Many binary tree operations are done by performing a traversal of thebinary 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, evaluatethe operator, etc.) with respect to this element is taken.
75 / 100
Images/cinvestav-1.jpg
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100
Images/cinvestav-1.jpg
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100
Images/cinvestav-1.jpg
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100
Images/cinvestav-1.jpg
Binary Tree Traversal Methods
Thus, we have
Preorder
Inorder
Postorder
Level order
76 / 100
Images/cinvestav-1.jpg
Remarks
First
In a traversal of a binary tree, each element of the binary tree is visitedexactly once.
Second
During the visit of an element, all action (make a clone, display, evaluatethe operator, etc.) with respect to this element is taken.
77 / 100
Images/cinvestav-1.jpg
Remarks
First
In a traversal of a binary tree, each element of the binary tree is visitedexactly once.
Second
During the visit of an element, all action (make a clone, display, evaluatethe operator, etc.) with respect to this element is taken.
77 / 100
Images/cinvestav-1.jpg
Preorder Traversals
Code
p u b l i c s t a t i c vo i d p reOrde r ( BinaryTreeNode t ){
i f ( t != n u l l ){
v i s i t ( t ) ;p reOrde r ( t . l e f t C h i l d ) ;p reOrde r ( t . r i g h t C h i l d ) ;
}}
78 / 100
Images/cinvestav-1.jpg
Example I : (visit = print)
Something Notable
79 / 100
Images/cinvestav-1.jpg
Example II : (visit = print)
Something Notable
80 / 100
Images/cinvestav-1.jpg
Actually the preorder can be used to express the pre�xform!!!
Example
81 / 100
Images/cinvestav-1.jpg
Inorder Traversals
Code
p u b l i c s t a t i c vo i d i nOrde r ( BinaryTreeNode t ){
i f ( t != n u l l ){
i nOrde r ( t . l e f t C h i l d ) ;v i s i t ( t ) ;i nOrde r ( t . r i g h t C h i l d ) ;
}}
82 / 100
Images/cinvestav-1.jpg
Example I : (visit = print)
Something Notable
83 / 100
Images/cinvestav-1.jpg
Example II : (visit = print)
Something Notable
84 / 100
Images/cinvestav-1.jpg
Example III : (visit = print)
Inorder By Projection (Squishing)
85 / 100
Images/cinvestav-1.jpg
Inorder Of Expression Tree
Example
Not only you get the in�x form
But, we have a implicit representation of the parenthesis
86 / 100
Images/cinvestav-1.jpg
Inorder Of Expression Tree
Example
Not only you get the in�x form
But, we have a implicit representation of the parenthesis
86 / 100
Images/cinvestav-1.jpg
Postorder Traversal
Code
p u b l i c s t a t i c vo i d pos tOrde r ( BinaryTreeNode t ){
i f ( t != n u l l ){
pos tOrde r ( t . l e f t C h i l d ) ;pos tOrde r ( t . r i g h t C h i l d ) ;v i s i t ( t ) ;
}}
87 / 100
Images/cinvestav-1.jpg
Example I : (visit = print)
Something Notable
88 / 100
Images/cinvestav-1.jpg
Example II : (visit = print)
Something Notable
89 / 100
Images/cinvestav-1.jpg
Actually the preorder can be used to express the post�xform!!!
Example
90 / 100
Images/cinvestav-1.jpg
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100
Images/cinvestav-1.jpg
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100
Images/cinvestav-1.jpg
Traversal Applications
First One
Make a Clone
Second
Determine the height.
Third
Determine number of nodes.
91 / 100
Images/cinvestav-1.jpg
Now
What if we want to traverse each node in every level �rst - level order
92 / 100
Images/cinvestav-1.jpg
Level Order
Code
Which one?
The code
93 / 100
Images/cinvestav-1.jpg
Level Order
Code
Which one?
The code
Let t be the t r e e r oo t .wh i l e ( t != n u l l ){
v i s i t t and put i t s c h i l d r e n on a FIFO queue ;remove a node from the 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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
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
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
Preorder And Postorder
If we have the following
Something Notable
Preorder = ab
Postorder = ba
We have a problem
Preorder and postorder do not uniquely de�ne a binary tree.
Nor do preorder and level order (same example).
Nor do postorder and level order (same example).
95 / 100
Images/cinvestav-1.jpg
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 rightsubtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the rightsubtree.
96 / 100
Images/cinvestav-1.jpg
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 rightsubtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the rightsubtree.
96 / 100
Images/cinvestav-1.jpg
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 rightsubtrees.
Thus
a is the root of the tree; gdhbei are in the left subtree; fjc are in the rightsubtree.
96 / 100
Images/cinvestav-1.jpg
Then
First
preorder = a b d g h e i c f j
97 / 100
Images/cinvestav-1.jpg
Then
First
preorder = a b d g h e i c f j
97 / 100
Images/cinvestav-1.jpg
Cont...
d is the next root; g is in the left subtree; h is in the right subtree.
98 / 100
Images/cinvestav-1.jpg
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Postorder
Thus
Scan postorder from right to left using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and rightsubtrees.
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
Images/cinvestav-1.jpg
Inorder And Level Order
Thus
Scan level order from left to right using inorder to separate left and rightsubtrees.
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