1. Compiled by: Dr. Mohammad Omar Alhawarat
Chapter 06
Compiled by: Dr. Mohammad Omar Alhawarat
Trees

2. Definition of Tree
A tree is a set of linked nodes, such that there is one and only one path from a unique node (called the root node) to every other node in the tree

3. Hierarchical Organization
Example: File Directories

4. 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

5. Example of a Tree
root A C B D G E F

6. Example of a Tree
In a tree, every pair of linked nodes have a parent-child relationship (the parent is closer to the root)
root A C B D G E F

7. Example of a Tree (Cont.)
For example, C is a parent of G
root A C B D G E F

8. Example of a Tree (Cont.)
E and F are children of D
root A C B D G E F

9. Example of a Tree (Cont.)
The root node is the only node that has no parent.
root A C B D G E F

10. Example of a Tree (Cont.)
Leaf nodes (or leaves for short) have no children.
root A C B D G E F

11. Subtrees root A subtree is a part of a tree that is a tree in itself AC I K D E F J G H
subtree

12. Subtrees (Cont.)
root
It normally includes each node reachable from its root. A B C I K D E F J G H
subtree

13. Subtrees (Cont.) root Even though this looks like a subtree in itself…D E F J G H

14. Binary Trees
A binary tree is a tree in which each node can only have up to two children…

15. Not a Binary Tree
root A B C I K D E F J G H

16. Example of a Binary Tree
root A
The links in a tree are often called edges B C I K E J G H

17. Levels
The level of a node is the number of edges in the path from the root node to this node
root
level 0 A
level 1 B C
level 2 I K E
level 3 J G H

18. Full Binary Tree root A B C D E F G H I J K L M N O
In a full binary tree, each node has two children except for the nodes on the last level, which are leaf nodes

19. Complete Binary Trees
A complete binary tree is a binary tree that is either
a full binary tree OR
a tree that would be a full binary tree but it is missing the rightmost nodes on the last level

20. Complete Binary Trees (Cont.)
root A B C D E F G H I

21. Complete Binary Trees (Cont.)
root A B C D E F G H I J K L

22. Full Binary Trees A full binary tree is also a complete binary tree.
root A B C D E F G H I J K L M N O

23. Full Binary Trees (Cont.)
The formula for the maximum number of nodes is derived from the fact that each node can have only two descendants. Given a height of the binary tree, H, the maximum number of nodes in the full binary tree is given as follows:

24. Balanced Binary Trees
To Assure that a binary tree is Balanced one the following algorithms is used:
AVL Trees.
Red-Black Trees.
2-3 Trees and the general case (B-Trees) B-Trees are used with Fetching data from Large Databases.

25. Binary Trees A binary tree is either empty or has the following form
Where Tleft and Tright are binary trees

26. 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)

27. Binary Trees

28. Recursive definition of a tree
A tree is a set of nodes that either:
is empty or
has a designated node, called the root, from which hierarchically descend zero or more subtrees, which are also trees.

29. Recursive definition of a tree (Cont.)

30. Tree Traversal
Four meaningful orders in which to traverse a binary tree.
Preorder
Inorder
Postorder
Level order

31. Tree Traversal (Cont.)

32. Tree Traversal (Cont.) In Preorder, the root is visited before (pre)
the subtrees traversals
In Inorder, the root is
visited in-between left
and right subtree traversal
In Postorder, the root
is visited after (post)
Preorder Traversal:
Visit the root
Traverse left subtree
Traverse right subtree
Inorder Traversal:
Traverse left subtree
Visit the root
Traverse right subtree
Postorder Traversal:
Traverse left subtree
Traverse right subtree
Visit the root

33. Tree Traversal (Cont.) Visiting a node
Processing the data within a node
This is the action performed on each node during traversal of a tree
A traversal can pass through a node without visiting it at that moment
For a binary tree
Visit the root
Visit all nodes in the root's left subtree
Visit all nodes in the root's right subtree

34. Tree Traversal (Cont.)
Preorder traversal: visit root before the subtrees

35. Tree Traversal (Cont.)
Inorder traversal: visit root between visiting the subtrees

36. Tree Traversal (Cont.)
Postorder traversal: visit root after visiting the subtrees
These are examples of a depth-first traversal.

37. Tree Traversal (Cont.)
Level-order traversal: begin at the root, visit nodes one level at a time
This is an example of a breadth-first traversal.

38. Tree Traversals – Example 1

39. Tree Traversals – Example 2
Assume: visiting a node
is printing its label
Preorder:
Inorder:
Postorder: 1 3 11 9 8 4 6 5 7 12 10

40. Tree Traversals – Example 3
Assume: visiting a node
is printing its data
Preorder:
Inorder:
Postorder: 6 15 8 2 3 7 11 10 14 12 20 27 22 30

41. Tree Traversals – Example 4

42. Examples of Binary Trees
Expression Trees

43. Example: Expression Trees
Cpt S 223
Example: Expression Trees
Store expressions in a binary tree
Leaves of tree are operands (e.g., constants, variables)
Other internal nodes are unary or binary operators
Used by compilers to parse and evaluate expressions
Arithmetic, logic, etc.
E.g., (a + b * c)+((d * e + f) * g) 43 43 43
Washington State University

44. Tree Traversal - Summary
Level order traversal is sometimes called breadth- first.
The other traversals are called depth-first.
Traversal takes Θ(n) in both breadth-first and depth- first.
Memory usage in a perfect tree is Θ(log n) in depth- first and Θ(n) in breadth-first traversal.

45. Questions?
Questions ?