1. Tonga Institute of Higher Education
Binary Trees
Tonga Institute of Higher Education

2. Review of Ordered Arrays and Linked Lists
Searching is fast.
Binary Search
O(logN)
Inserting and Deleting is slow.
Need to move many values one space over
O(N)
Linked List
Inserting and deleting is fast.
Just need to change a few references
O(1)
Searching is slow
Must start from the beginning of the list and check every link.

3. Introduction to Binary Trees
A Binary Tree combines the advantages of ordered arrays and linked lists
Like an ordered array, we can:
Search for an item quickly
Like a linked list, we can:
Insert items quickly
Delete items quickly

4. What is a Tree? A tree consists of nodes connected by edges
Nodes represent objects
Example: Customers, plane tickets, etc.
Edges represent references
There is 1 node at the top of the tree
Trees are small on top and large on bottom
It looks like an upside down tree

5. Tree Terminology - 1 Path – The route you must take to get to a node
Root – The top node of the tree
Parent – Every node must have one edge running upwards to another node. The node above it is a parent
Child – Any node may have one or more edges running downward to another node. The node below it is a child.

6. Tree Terminology - 2 Leaf – A node with no children
Subtree – Any node may be considered to be the root of a subtree, which consists off all the edges and nodes under that node.
Visiting – A node is visited when a program control arrives at a node to carry out an operation like checking a value. Just passing over a node on a path from one node to another is not considered to be visiting the node.

7. Tree Terminology - 3
Traversing – To traverse a tree means to visit all nodes in a specified order.
Levels – The level of a node refers to how many edges the node is from the root.
Keys – One data field in an object stored in a node usually contains a key value. This value is used to search for the item or perform other operations on it.

8. Tree Rules
This is not
a valid tree
because there
is a node with
more than one
path to the root
The only way to move from one node to another is to follow a path along the lines
There must be only one path to the root.

9. Using Trees in Everyday Life
You have been using a Tree for at least half a year already!
The file structure in your computer follows these rules!
Desktop is the root node
Directories are nodes
Directories without any subdirectories or files in them are leaves
Files are leaves
You double-click on a folder to move from one node to another

10. Binary Trees Binary Trees contain number keys
Binary Trees can only have a maximum of 2 children
The key in the left child node must be less than or equal to the key in the right child node

11. Presentation
Binary Tree Applet

12. Node Class We can create an class to represent a node
The class can contain data representing the object being stored
Employees
Items
The class must also contain
A key value
A left node
A right node

13. Code View
Node Class

14. Tree Class We can create an class to represent a tree
A tree has only one field: a node that holds the root
The tree class can have many methods
Find
Insert
Delete
Etc.

15. Code View
Tree Class

16. Finding a Node Each node can represent an object Each object has a key
Customer
Item
Each object has a key
Customer ID
Item ID
We can find an item easily with these steps
At a node, determine whether the key of the object you are looking for is greater than, equal to, or less than the value of the node
If the key is less: Move to the left child node and repeat from step 1.
If no left child node is available, the node does not exist in this tree.
If the key is equal: We have found the node we are looking for
If the key is greater: Move to the right child node and repeat from step 1.
If no right child node is available, the node does not exist in this tree.

17. Demonstration
Find in Applet

18. Code View
Find Code

19. Find Efficiency
The time required to find a node depends on the level that the node is located
This is O(logN) time

20. Inserting a Node We can insert an item with these steps
Find the place to put the node
Pretend like we’re looking for a node
When we get to the last node and realize that the node we want to insert does not already exist, we add it as a left or right child node

21. Unbalanced Trees
Sometimes, when we add a lot of nodes, our tree becomes very heavy on one side
This condition is called an unbalance tree
How to handle unbalanced trees is covered in Algorithm Analysis and Design

22. Demonstration
Insert in Applet

23. Code View
Insert Code

24. Traversing a Tree
Traversing – To traverse a tree means to visit all nodes in a specified order.
There are 3 ways to traverse a tree
Inorder
Preorder
Postorder

25. Inorder Traversal - 1
An inorder traversal of a binary search tree will cause all the nodes to be visited in ascending order.
To traverse a tree Inorder we repeat the 3 following steps until we have visited every node in the binary tree
Go to left child
Visit the node
Go to right child

26. Inorder Traversal - 2 Start from the root node Move to right child
left child

27. Inorder, Preorder and Postorder Traversals
Inorder Traversal
Go to left child
Visit the node
Go to right child
Preorder Traversal
Postorder Traversal

28. Demonstration
Traversals in Applet

29. Code View
Traversal Code

30. Finding Maximum and Minimum Values
To get the minimum value, go to the left-most node
To get the maximum value, go to the right-most node

31. Deleting a Node with No Children
To delete a node with no children, set the reference of the parent node to be null.
The node will eventually be cleaned up by the Java garbage collector.

32. Deleting a Node with 1 Child
To delete a node with 1 child, set the reference of the parent node to be the child of the deleted node.

33. Deleting a Node with 2 Children - 1
When you are deleting a node with 2 children, we have a problem.
How do we know which node to put in the place of the deleted node?
In other words, which node is the successor?

34. Deleting a Node with 2 Children - 2
Subset of the
nodes on the
right
To find the successor we must find the smallest value of the subset of the nodes on the right
To do this:
Go to the right child of the node to be deleted
Then, find the leftmost node

35. Finding a Successor Node
These nodes are
the leftmost nodes
of the right child!
In this case the successor node is
the right child of the node to be deleted
In this case, the successor node is
not the right child of the node to be deleted

36. Deleting a Node when the Successor is the Right Child of the Node to be Deleted
87 replaces 75
because it was
the successor
When the successor is the right child of the node to be deleted, replace the deleted node with the successor

37. Deleting a Node when the Successor is a Left Descendant of Right Child Node of the Node to be Deleted
The successor replaces the deleted node
The algorithm is redone as if the successor was deleted until the entire tree has been redone

38. Demonstration
Deletions in Applet

39. Code View
Deletion Code

40. Binary Tree Summary with Driver
Code View
Binary Tree Summary with Driver