1. Binary Trees
Lecture 36
Wed, Apr 21, 2004
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2. Topics Binary trees Array implementation Linked implementation
Binary tree nodes
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3. Binary Trees
A binary tree is a data structure with the following properties.
It is either empty or it has a root node.
Each node in the binary tree may have up to two children (called left and right).
Each node, except the root node, has exactly one parent. The root node has no parent.
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4. A Binary Tree 10 20 30 40 50 60 70
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5. Terminology The tree metaphor The family metaphor
tree, root, branch, leaf.
The family metaphor
parent, child, sibling, ancestor, descendant.
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6. Tree Terminology 10 20 30 40 50 60 70
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7. Tree Terminology
Root 10 20 30 40 50 60 70
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8. Tree Terminology 10 20 30 40 50 60 70
Parent
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9. Tree Terminology 10 20 30 40 50 60 70
Leaves
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10. Tree Terminology 10 20 30 40 50 60 70 Right Child 9/21/2018
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11. Tree Terminology
Subtree 10 20 30 40 50 60 70
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12. Array Implementation of a Binary Tree
In an array binary tree, the nodes of the tree are stored in an array.
Position 0 is left empty.
The root is stored in position 1.
For the element in position n,
The left child is in position 2n.
The right child is in position 2n + 1.
The parent is in position n/2.
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13. Array Implementation 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9 9/21/2018
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14. Array Implementation Unused 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9
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15. Array Implementation Root 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9
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16. Array Implementation Parent 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9
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17. Array Implementation Parents, do you know where your children are?10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9
Parents, do you know where your children are?
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18. Array Implementation Parents, do you know where your children are?10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9
Parents, do you know where your children are?
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19. Advantages of the Array Implementation
This representation is very efficient when
The tree is full, and
The structure of the tree will not be modified.
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20. Linked Implementation of a Binary Tree
In a linked binary tree, each node contains pointers to its left and right children.
A linked binary tree object has one data member:
BinaryTreeNode* root - A pointer to the root node.
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21. Binary Tree Nodes A binary tree node has three components.
A value (data) that contains the tree element.
A left pointer (left) that points to the root node of the left subtree.
A right pointer (right) that points to the root node of the right subtree.
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22. Binary Tree Nodes Nodes are allocated and deallocated dynamically.
The binary tree always has allocated the exact number of nodes necessary to store the tree elements.
Example
binarytreenode.h
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23. Binary Tree Implementation
Lecture 37
Thu, Apr 22, 2004
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24. Topics Binary tree interface Implementation of Search()
Implementation of Draw()
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25. Binary Tree Constructors
BinaryTree(const T& value);
BinaryTree(const BinaryTree& lft,
const BinaryTree& rgt);
BinaryTree(const T& value,
const BinaryTree& lft,
BinaryTree(const BinaryTree& tree);
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26. Binary Tree Destructor
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27. Binary Tree Inspectors
int Size() const;
int Height() const;
bool Empty() const;
T& Root() const;
BinaryTree LeftSubtree() const;
BinaryTree RightSubtree() const;
bool IsCountBalanced() const;
bool IsHeightBalanced() const;
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28. Binary Tree Mutators void MakeEmpty(); void SetRoot(const T& value);
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29. Binary Tree Facilitators
void Input(istream& in);
void Output(ostream& out);
bool Equal(BinaryTree tree) const;
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30. Binary Tree Operators BinaryTree& operator=(const BinaryTree&);
istream& operator>>(istream&,
BinaryTree&);
ostream& operator<<(ostream&,
const BinaryTree&);
bool operator==(BinaryTree&);
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31. Binary Tree Traversal Functions
void PreorderTraversal(void (* Visit)
(BinaryTreeNode*)) const;
void InorderTraversal(void (* Visit)
void PostorderTraversal(void (* Visit)
void LevelorderTraversal(void (* Visit)
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32. Other Binary Tree Functions
T* Search(const T& value) const;
void Draw() const;
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33. Linked Binary Tree Implementation
Example
binarytree.h
BinaryTreeTest.cpp
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34. Binary Tree Traversals
Lecture 38
Mon, Apr 26, 2004
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35. Topics Binary tree traversals Binary tree iterators
Pre-order traversals
In-order traversals
Post-order traversals
Level-order traversals
Binary tree iterators
Pre-order iterators
In-order iterators
Post-order iterators
Level-order iterators
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36. Binary Tree Traversals
Four Standard Traversals
Pre-order Traversal.
In-order Traversal.
Post-order Traversal.
Level-order Traversal.
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37. Pre-order Traversal At each node, Also called an NLR traversal.
First visit the node,
Then perform a pre-order traversal of the left subtree,
Then perform a pre-order traversal of the right subtree.
Also called an NLR traversal.
Node - Left - Right.
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38. Pre-Order Traversal 10 20 30 40 50 60 70
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39. Pre-Order Traversal 10 20 30 40 50 60 70 10
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40. Pre-Order Traversal 10 20 30 40 50 60 70 10 20
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45. Pre-Order Traversal 10 20 30 40 50 60 70 10 20 30 40 50 60 70
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46. Pre-Order Traversal Order: 10, 20, 40, 50, 30, 60, 70 10 20 30 40 50
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47. In-order Traversal At each node, Also called an LNR traversal.
First, perform an in-order traversal of the left subtree,
Then visit the node,
Then perform an in-order traversal of the right subtree.
Also called an LNR traversal.
Left - Node - Right.
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48. In-Order Traversal 10 20 30 40 50 60 70
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49. In-Order Traversal 10 20 30 40 50 60 70 40
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50. In-Order Traversal 10 20 30 40 50 60 70 20 40
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54. In-Order Traversal 10 20 30 40 50 60 70 10 20 30 40 50 60
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55. In-Order Traversal 10 20 30 40 50 60 70 10 20 30 40 50 60 70
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56. In-Order Traversal Order: 40, 20, 50, 10, 60, 30, 70 10 20 30 40 50 60
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57. Post-order Traversal At each node, Also called an LRN traversal.
First perform a post-order traversal of the left subtree,
Then perform a post-order traversal of the right subtree,
Then visit the node.
Also called an LRN traversal.
Left - Right - Node.
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58. Post-Order Traversal 10 20 30 40 50 60 70
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59. Post-Order Traversal 10 20 30 40 50 60 70 40
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60. Post-Order Traversal 10 20 30 40 50 60 70 40 50
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63. Post-Order Traversal 10 20 30 40 50 60 70 20 40 50 60 70 9/21/2018
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64. Post-Order Traversal 10 20 30 40 50 60 70 20 30 40 50 60 70 9/21/2018
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65. Post-Order Traversal 10 20 30 40 50 60 70 10 20 30 40 50 60 70
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66. Post-Order Traversal Order: 40, 50, 20, 60, 70, 30, 10 10 20 30 40 50
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67. Level-order Traversal
Traverse the levels of the tree from top to bottom.
Within each level, visit the nodes from left to right.
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68. Level-Order Traversal10 20 30 40 50 60 70
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69. Level-Order Traversal10 20 30 40 50 60 70 10
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70. Level-Order Traversal10 20 30 40 50 60 70 10 20
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71. Level-Order Traversal10 20 30 40 50 60 70 10 20 30
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72. Level-Order Traversal10 20 30 40 50 60 70 10 20 30 40
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73. Level-Order Traversal10 20 30 40 50 60 70 10 20 30 40 50
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74. Level-Order Traversal10 20 30 40 50 60 70 10 20 30 40 50 60
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75. Level-Order Traversal10 20 30 40 50 60 70 10 20 30 40 50 60 70
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76. Level-Order Traversal10 20 30 40 50 60 70 10 20 30 40 50 60 70
Order: 10, 20, 30, 40, 50, 60, 70
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77. Binary Tree Iterators
A binary tree iterator systematically visits each node of a binary tree.
The four standard iterators correspond to the four standard traversal methods.
Pre-order iterator
In-order iterator
Post-order iterator
Level-order iterator
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78. Pre-order Iterator Visit the nodes in a pre-order traversal.
Use a stack to store unvisited nodes.
Example
binarytreeiterator.h
binarytreepreorderiterator.h
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79. Pre-order Iterator 10 20 30 40 50 60 70 10 10 10 20 20 20 30 30 30 40 50 60 70
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80. In-order Iterator Visit the nodes in an in-order traversal.
Use a stack to store unvisited nodes.
Example
binarytreeiterator.h
binarytreeinorderiterator.h
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81. In-order Iterator 10 20 30 40 50 60 70 10 10 20 20 30 30 40 50 60 70
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82. Post-order Iterator Visit the nodes in a post-order traversal.
Use a stack to store unvisited nodes.
Example
binarytreeiterator.h
binarytreepostorderiterator.h
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83. Post-order Iterator 10 20 30 40 50 60 70 10 10 20 20 30 30 40 50 60 70
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84. Level-order Iterator Visit the nodes in a level-order traversal.
Use a queue to store unvisited nodes.
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85. Level-order Iterator 10 20 30 40 50 60 70 10 20 20 30 30 40 40 50 50 60 60 70 70
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86. Traversals and Expression Trees
Perform an in-order traversal of the expression tree and print the nodes. * + - 4 5 6 3
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87. Traversals and Expression Trees
Perform a post-order traversal to evaluate the expression tree. * + - 4 5 6 3
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88. Binary Search Trees
Lecture 39
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89. Topics Binary search trees Inserting a node Deleting a node
Balancing a binary search tree
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90. Binary Search Trees
A binary search tree is a binary tree with the following properties.
There is a total order relation on the members in the tree.
At every node, every member of the left subtree is less than or equal to the node value.
At every node, every member of the right subtree is greater than or equal to the node value.
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91. BinarySearchTree Implementation
The BinarySearchTree class is implemented as a subclass of the BinaryTree class.
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92. Binary Search Tree Interface
Mutators
void Insert(const T& value);
void Delete(const T& value);
Other member functions
T* Search(const T& value) const;
void CountBalance();
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93. Searching a BinarySearchTree
Begin at the root node.
Apply the following algorithm recursively.
Compare the value to the node data.
If it is equal, you are done.
If it is less, search the left subtree.
If it is greater, search the right subtree.
If the subtree is empty, the value is not in the tree.
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94. Inserting a Value into a BinarySearchTree
Begin at the root node.
Apply the following algorithm recursively.
Compare the value to the node data.
If it is less (or equal), continue with the left subtree.
If is is greater, continue with the right subtree.
When the subtree is empty, attach the node as a subtree.
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95. Deleting a Value from a BinarySearchTree
Perform a search to locate the value.
This node has
Two children.
One child.
No child.
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96. Deleting a Value from a BinarySearchTree
Case 1: The node has no child.
Delete the node.
Case 2: The node has one child.
Replace the node with the subtree of which the child is the root.
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97. Deleting a Value from a BinarySearchTree
Case 3: The node has two children.
Locate the next smaller value in the tree.
This value is the rightmost value of the left subtree.
Move left one step.
Move right as far as possible.
Swap this value with the value to be deleted.
The node to be deleted now has at most one child.
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98. Count-Balancing a BinarySearchTree
Write a function MoveNodeRight() that will move the largest value of the left subtree to the right subtree.
Locate the largest value in the left subtree.
Delete it (but save the value).
Place it at the root.
Insert the old root value into the right subtree.
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99. Count-Balancing a BinarySearchTree
Write a similar function MoveNodeLeft().
Apply either MoveNodeRight() or MoveNodeLeft() repeatedly at the root node until the tree is balanced at the root.
Then apply these functions recursively, down to the leaves.
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100. BinarySearchTree Implementation
Example
binarysearchtree.h
BinarySearchTreeTest.cpp
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