1. การวิเคราะห์และออกแบบขั้นตอนวิธี
CSE 221
การวิเคราะห์และออกแบบขั้นตอนวิธี
Lecture 07: การวิเคราะห์ขั้นตอนวิธีที่ใช้ในโครงสร้าง
ข้อมูลแบบ Trees
ดร.สุรศักดิ์ มังสิงห์
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2. Trees
Trees are a very useful data structure. Many different kinds of trees are used in Computer Science. We shall study just a few of these.
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3. Introduction to Trees General Trees Binary Trees Binary Search Trees
AVL Trees
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4. Tree
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5. Definition A tree t is a finite nonempty set of elements.
One of these elements is called the root.
The remaining elements, if any, are partitioned into trees, which are called the subtrees of t.
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6. Sub-trees
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7. Tree
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8. height = depth = number of levels
Object
Number
Throwable
OutputStream
Integer
Double
Exception
FileOutputStream
RuntimeException
Level 4
Level 2
Level 1
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9. Node Degree = Number Of Children3
Object
Number
Throwable
OutputStream
Integer
Double
Exception
FileOutputStream
RuntimeException 1 2 1 1
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10. Tree Degree = Max Node Degree3
Object
Number
Throwable
OutputStream
Integer
Double
Exception
FileOutputStream
RuntimeException 1 2 1 1
Note that it is possible to have a tree whose degree is (say) 3 and whose root has a
degree that is < 3.
Degree of tree = 3.
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11. Binary Tree
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12. Binary Tree Finite (possibly empty) collection of elements.
A nonempty binary tree has a root element.
The remaining elements (if any) are partitioned into two binary trees.
These are called the left and right subtrees of the binary tree.
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13. Binary Tree
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14. A Tree vs A Binary Tree
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.
A binary tree may be empty; a tree cannot be empty.
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15. Are different when viewed as binary trees.
A Tree vs A Binary Tree
The subtrees of a binary tree are ordered; those of a tree are not ordered. a b
Are different when viewed as binary trees.
Are the same when viewed as trees.
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16. Forms of Binary Trees
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17. Complete Binary Trees
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18. Tree Traversal
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19. Processing and Walking Order
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20. Depth First Processing
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21. Preorder Traversal
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22. Breath First Processing
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23. Height and number of nodes
Maximum height of a binary tree
Hmax = N
Minimum height of a binary tree
Hmin = logN + 1
Maximum and Minimum number of nodes
Nmin = H and Nmax = 2H - 1
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24. 5/18/2019

25. การประยุกต์ใช้ Tree
Expression Tree
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26. Arithmetic Expressions
(a + b) * (c + d) + e – f/g*h
Expressions comprise three kinds of entities.
Operators (+, -, /, *).
Operands (a, b, c, d, e, f, g, h, 3.25, (a + b), (c + d), etc.).
Delimiters ((, )).
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27. Infix Form Normal way to write an expression.
Binary operators come in between their left and right operands.
a * b
a + b * c
a * b / c
(a + b) * (c + d) + e – f/g*h
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28. Operator Priorities How do you figure out the operands of an operator?
a + b * c
a * b + c / d
This is done by assigning operator priorities.
priority(*) = priority(/) > priority(+) = priority(-)
When an operand lies between two operators, the operand associates with the operator that has higher priority.
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29. Tie Breaker
When an operand lies between two operators that have the same priority, the operand associates with the operator on the left.
a + b - c
a * b / c / d
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30. Delimiters
Subexpression within delimiters is treated as a single operand, independent from the remainder of the expression.
(a + b) * (c – d) / (e – f)
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31. Infix Expression Is Hard To Parse
Need operator priorities, tie breaker, and delimiters.
This makes computer evaluation more difficult than is necessary.
Postfix and prefix 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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32. Postfix Form
The postfix form of a variable or constant is the same as its infix form.
a, b, 3.25
The relative order of operands is the same in infix and postfix forms.
Operators come immediately after the postfix form of their operands.
Infix = a + b
Postfix = ab+
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33. Postfix Examples a b c * + a b * c + a b + c d - * e f + /
Infix = a + b * c
Postfix = a b c * +
Infix = a * b + c
Postfix = a b * c +
Infix = (a + b) * (c – d) / (e + f)
Postfix = a b + c d - * e f + /
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34. Expression Tree
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35. Expression Tree
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36. Binary Tree Form + a b - a - a a + b
Each leaf represents a variable or constant. Each nonleaf represents an operator. The left operand (if any) of this operator is represented by the left subtree, the right subtree represents the right operand of the operator.
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37. Binary Tree Form / + a b - c d e f * (a + b) * (c – d) / (e + f)
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38. Expression Tree
Infix Expression =?
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39. Constructing an Expression Tree
a b + c d * -
(a)
(b) a a b + c b d
(c)
(d) + * - a b c d + * a b c d
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40. การประยุกต์ใช้ Tree
Binary Search Trees
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41. Figure 8-1
Binary Search Tree
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42. Binary Search Trees
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43. Are these Binary Search Trees?
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44. Construct a Binary Search Tree
เวลาที่ใช้ในการค้นหาข้อมูล Worst case? Average case?
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45. 5/18/2019

46. Balance Binary Search Tree
AVL Trees
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47. AVL Trees
Balanced binary tree structure, named after Adelson, Velski, and Landis
An AVL tree is a height balanced binary search tree.
|HL – HR| <= 1
where HL is the height of the left subtree and HR is the height of the left subtree
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48. Binary Search Trees
(b) AVL Tree
(a) An unbalanced BST
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49. Out of Balance Four cases of out of balance:
left of left - requires single rotation
right of right - requires single rotation
Left of right - requires double rotation
Right of left - requires double rotation
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50. Out of Balance (left of left)
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51. Out of Balance (left of left)
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52. Out of Balance (right of right)
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53. Out of Balance (right of right)
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54. Simple double rotation right
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55. Complex double rotation right
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56. Insert a node to AVL tree
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57. Balancing BST
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58. Deleting a node from AVL tree
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59. Balance Binary Search Tree
เวลาที่ใช้ในการค้นหาข้อมูลใน AVL Tree Worst case? Average case?
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60. 5/18/2019

61. Priority Queue
Heap Sort
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62. Min Priority Queue Collection of elements.
Each element has a priority or key.
Supports following operations:
isEmpty
size
add/put an element into the priority queue
get element with min priority
remove element with min priority

63. Max Priority Queue Collection of elements.
Each element has a priority or key.
Supports following operations:
isEmpty
size
add/put an element into the priority queue
get element with max priority
remove element with max priority

64. Root has minimum element.
Min Tree Example 2 4 9 3 4 8 7 9 9
Root has minimum element.

65. Root has maximum element.
Max Tree Example 9 4 9 8 4 2 7 3 1
Root has maximum element.

66. complete binary tree min tree
Min Heap Definition
complete binary tree
min tree

67. Complete binary tree with 9 nodes that is also a min tree.
Min Heap With 9 Nodes 2 4 6 7 9 3 8
Complete binary tree with 9 nodes that is also a min tree.

68. Complete binary tree with 9 nodes that is also a max tree.
Max Heap With 9 Nodes 9 8 6 7 2 5 1
Complete binary tree with 9 nodes that is also a max tree.

69. What is the height of an n node heap ?
Heap Height
What is the height of an n node heap ?
Since a heap is a complete binary tree, the height of an n node heap is log2 (n+1).
Since height must be an integer, the precise expression for height is ceiling(log2 (n+1)).

70. A Heap Is Efficiently Represented As An Array9 8 6 7 2 5 1 9 8 7 6 2 5 1 3 4 10

71. Moving Up And Down A Heap9 8 6 7 2 5 1 3 4

72. Putting An Element Into A Max Heap9 8 6 7 2 5 1 7
Complete binary tree with 10 nodes.

73. Putting An Element Into A Max Heap9 8 7 6 7 2 6 5 1 5 7
New element is 5.

74. Putting An Element Into A Max Heap9 8 7 6 7 2 6 5 1 7 7
New element is 20.

75. Putting An Element Into A Max Heap9 8 7 6 2 6 5 1 7 7 7
New element is 20.

76. Putting An Element Into A Max Heap9 7 6 8 2 6 5 1 7 7 7
New element is 20.

77. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
New element is 20.

78. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
Complete binary tree with 11 nodes.

79. Putting An Element Into A Max Heap20 9 7 6 8 2 6 5 1 7 7 7
New element is 15.

80. Putting An Element Into A Max Heap20 9 7 6 2 6 5 1 7 7 8 7 8
New element is 15.

81. Putting An Element Into A Max Heap20 15 7 6 9 2 6 5 1 7 7 8 7 8
New element is 15.

82. Complexity is O(log n), where n is heap size.
Complexity Of Put 8 6 7 2 5 1 20 9 15
Complexity is O(log n), where n is heap size.

83. Removing The Max Element8 6 7 2 5 1 20 9 15
Max element is in the root.

84. Removing The Max Element15 7 6 9 2 6 5 1 7 7 8 7 8
After max element is removed.

85. Removing The Max Element15 7 6 9 2 6 5 1 7 7 8 7 8
Heap with 10 nodes.
Reinsert 8 into the heap.

86. Removing The Max Element15 7 6 9 2 6 5 1 7 7 7
Reinsert 8 into the heap.

87. Removing The Max Element15 7 6 9 2 6 5 1 7 7 7
Reinsert 8 into the heap.

88. Removing The Max Element15 9 7 6 8 2 6 5 1 7 7 7
Reinsert 8 into the heap.

89. Removing The Max Element15 9 7 6 8 2 6 5 1 7 7 7
Max element is 15.

90. Removing The Max Element9 7 6 8 2 6 5 1 7 7 7
After max element is removed.

91. Removing The Max Element9 7 6 8 2 6 5 1 7 7 7
Heap with 9 nodes.

92. Removing The Max Element9 7 6 8 2 6 5 1
Reinsert 7.

93. Removing The Max Element9 7 6 8 2 6 5 1
Reinsert 7.

94. Removing The Max Element9 8 7 6 7 2 6 5 1
Reinsert 7.

95. Complexity Of Remove Max Element6 2 5 1 7 9 8
Complexity is O(log n).

96. Complexity Of Operations
Two good implementations are heaps and leftist trees.
isEmpty, size, and get => O(1) time
put and remove => O(log n) time where n is the size of the priority queue

97. การประยุกต์ใช้ Tree กับปัญหาต้องตัดสินใจ
Next Lecture
การประยุกต์ใช้ Tree กับปัญหาต้องตัดสินใจ
5/18/2019