1. Fundamentals of Programming II Introduction to Trees
Computer Science 112
Fundamentals of Programming II
Introduction to Trees

2. Categories of Collections
Unordered (bags, sets, dictionaries)
Linear (stacks, queues, lists)
Hierarchical (trees)
Graphs

3. Why Trees? Organization charts Decision flow charts
Logarithmic searching
Language processing

4. What Is a Tree? Each item has at most one predecessor
Each item can have many successors

5. What Is a Tree?
Root
node
Start with a single node, called the root

6. What Is a Tree? Start with a single node, called the root
Each successor node is a child or daughter node
Daughters of root node

7. What Is a Tree? Successors are also called descendants Root node
Descendants of root node

8. What Is a Tree? Nodes without successors are called leaf nodes
Root
node
Nodes without successors are called leaf nodes
The set of leaf nodes is the frontier
Leaf nodes (the frontier)

9. What Is a Tree?
Root
node
Nodes with at least one successor are called interior nodes
Interior nodes

10. What Is a Tree? Predecessors are also called ancestors
Root
node
Predecessors are also called ancestors
The immediate predecessor is called the parent X
Ancestors of node X

11. What Is a Tree? Nodes with the same parent are called siblings Root

12. What Is a Tree? Levels in a tree are numbered from 0 Root node Level 0
There are three nodes in level 3

13. What Is a Tree? A binary tree allows at most two successors per node
Root
node
A binary tree allows at most two successors per node

14. What Is a Tree?
Root
node
A general tree allows any number of successors per node

15. Trees as Recursive Data Structures
A tree is either
empty, or
consists of a node containing
a datum
one or more subtrees
Each subtree is itself another tree

16. Tree Traversals We’d like to visit each data item in a tree
Are the items randomly ordered, as in a bag or set?
Think of visiting the data in a node, and its left and right subtrees, in some order

17. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited: D B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

18. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

19. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B A B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

20. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B A C B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

21. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B A C F B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

22. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B A C F E B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

23. Preorder Traversal Visit the data Visit the left subtree
Order of nodes visited:
D B A C F E G B F A C E G
Visit the data
Visit the left subtree
Visit the right subtree

24. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited: A B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

25. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

26. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B C B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

27. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B C D B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

28. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B C D E B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

29. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B C D E F B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

30. Inorder Traversal Visit the left subtree Visit the data
Order of nodes visited:
A B C D E F G B F A C E G
Visit the left subtree
Visit the data
Visit the right subtree

31. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited: A B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

32. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

33. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

34. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B E B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

35. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B E G B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

36. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B E G F B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

37. Postorder Traversal Visit the left subtree Visit the right subtree
Order of nodes visited:
A C B E G F D B F A C E G
Visit the left subtree
Visit the right subtree
Visit the data

38. Level Order Traversal For each level Visit data from left to right D
Order of nodes visited:
D B F A C E G B F A C E G
For each level
Visit data from left to right

39. Tree Applications File directories
Processing sentences (computer programs or natural languages)
Searchable data structures
Heaps (implement heap sort, priority queues)

40. For Monday
Binary Search Trees