This document discusses trees and binary trees. It covers tree terminology, tree traversals including preorder, inorder and postorder, implementing binary trees in Java, using binary trees to represent general trees, and applications of trees for data science. The next lecture will cover binary search trees, AVL trees, binary heaps and red black 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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