Data Structures and Algorithms Week 4 Trees Binary Trees

1. Data Structures and
Algorithms
Week 4: Trees
Ferdin Joe John Joseph, PhD
Faculty of Information Technology
Thai-Nichi Institute of Technology, Bangkok

2. Week 4
• Tree ADT and its applications
• Tree Traversals
• Binary Trees
• Implementation in Java
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3. Tree ADT
• Example: Family trees
Carole's children and grandchildren.
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4. Hierarchical Organization
• Example: Family trees
Jared's parents and grandparents.
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5. Hierarchical Organization
• Example: A university's organization
A university's administrative structure.
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6. Hierarchical Organization
• Example: File Directories
Computer files organized into folders
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7. Tree Terminology
• A tree is
• A set of nodes
• Connected by edges
• The edges indicate relationships among nodes
• Nodes arranged in levels
• Indicate the nodes' hierarchy
• Top level is a single node called the root
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8. Tree Terminology
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9. Tree Terminology
• Nodes at a given level are children of nodes of
previous level
• Node with children is the parent node of those
children
• Nodes with same parent are siblings
• Node with no children is a leaf node
• The only node with no parent is the root node
• All others have one parent each
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10. Tree Terminology
• Empty trees?
• Some authors specify a general tree must have at least
the root node
• This text will allow all trees to be empty
• A node is reached from the root by a path
• The length of the path is the number of edges that
compose it
• The height of a tree is the number of levels in the
tree
• The subtree of a node is a tree rooted at a child of
that node
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11. Binary Trees
• Each node has at most two children
Three binary trees.
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12. • A binary tree is either empty or has the following
form
• Where Tleft and Tright are binary trees
Binary Trees
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13. Binary Trees
• Every nonleaf in a full binary tree has exactly two
children
• A complete binary tree is full to its next-to-last level
• Leaves on last level filled from left to right
• The height of a binary tree with n nodes that is
either complete or full is
log2(n + 1)
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14. Nodes in a Binary Tree
A node in a binary tree.
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15. Interface for a Node
• Usually the class that represents a node in a tree is
a detail hidden from the client
• It is placed within a package
• Makes it available
• Available to all classes involved in implementation of a
tree
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16. The Method privateSetTree
• Problem: previous implementation of
privateSetTree not sufficient
• Given: treeA.setTree(a, treeB, treeC);
• Now treeA shares nodes with treeB, treeC
• Changes to treeB also affect treeA
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17. The Method privateSetTree
• Possible solution
• Have privatSetTree copy the nodes in TreeB and
TreeC
• Now treeA is separate and distinct from treeB and
treeC
• Changes to either treeB or treeC do NOT affect treeA
• Note that copying nodes is expensive
• Other solutions are considered
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18. The Method privateSetTree
• Another possible solution for
treeA.setTree(a, treeB, treeC);
• Have privateSetTree first link the root node of
treeA to root nodes of treeB, treeC
• Then set treeB.root, treeC.root to null
This still has some
problems
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19. The Method privateSetTree
1. Create root node r for containing the data
2. If left subtree exists, not empty
• Attach root node to r as left child
3. If right subtree exists, not empty, distinct from left
subtree
• Attach root node r as right child
• If right, left subtrees are the same, attach copy of right
subtree to r instead
4. If left (right) subtree exists, different from the
invoking tree object
• Set its data field root to null
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20. Methods for Binary Tree
• getRootData
• isEmpty
• clear
• setRootData
• setRootNode
• getRootNode
• BinaryNodeInterface
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21. Computing Height, Counting Nodes
• Within BinaryTree – methods in context
• getHeight()
• getNumberOfNodes()
• Within BinaryNode – methods in context
• getHeight()
• getHeight(BinaryNode node)
• getNumberOfNodes()
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22. Traversals
• Make recursive method private
• Call from public method with no parameters
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23. Traversals
A binary tree.
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24. Traversals
Using a stack to perform an inorder
traversal of the binary tree
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25. Traversals
Using a stack to traverse the
binary tree in (a) preorder
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26. Traversals
Using a stack to traverse the binary
tree in (b) postorder.
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27. Traversals
Using a queue
to traverse the
binary tree in
level order.
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28. General Trees
A node for a general tree.
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29. Using a Binary Tree to Represent a General
Tree
A general tree
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30. Using a Binary Tree to Represent a General
Tree
an equivalent binary tree;
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31. Using a Binary Tree to Represent a General
Tree
a more conventional view
of the same binary tree.
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32. Traversals
• Traversals of general tree in a
• Preorder: A B E F C G H I D J
• Postorder: E F B G H I C J D A
• Level order: A B C D E F G H I J
• Traversals of binary tree in c
• Preorder: A B E F C G H I D J
• Postorder: F E I H G J D C B A
• Level order: A B E C F G D H J I
• Inorder: E F B G H I C J D A
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33. Data Scientists’ view
• To connect a series of data corresponding to a
hierarchy
• Manage data from social network
• Blockchain transaction data analysis
• Binary decision sequence management in robotics
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34. Activity - Graded
• Grading for 3 points
• Write a java code to perform preorder and post
order traversal using stack in a binary tree.
• Bonus points: 1 If the submission is done before
the end of 8 AM 9 Feb 2019.
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35. Next Week
Binary Search Trees
AVL Trees
Binary Heaps
Red Black Trees
Implementation in Java
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