1. Data Structures – Binary Tree

2. What is a tree?

3. Where do you see trees?
Ummm...outside
Family trees
File systems

4. Tree Terminology node – any element of the tree
root – the topmost node of the tree
parent – a node that has one or more nodes connected below it (children)
child – a node that has a connected node above it (parent)
leaf – any child node at the bottom of the tree
subtree – any node and all its descendants

5. Name that part! root / parent subtree parent / child’ child / leaf

6. So what’s a binary tree?
A parent can have at most TWO children (left child, right child)
The left child’s data and all the data in the left subtree is “less than” the parent’s data
The right child’s data all the data in the right subtree is “greater than” the parent’s data
NOTE: The “less than” and “greater than” requirements of the subtrees is commonly known outside of the IB world as a Binary Search Tree

7. Example – Valid Binary Tree6 2 10 4 1

8. Example – INVALID Binary Tree3 2 10 4 1

9. Adding to a Binary Tree Start at the root
Compare against the current node
Go left if less than current node OR insert if there is none (creating a new leaf)
Go right if greater than current node OR insert if there is none (creating a new leaf)
Repeat step 2

10. Creating a binary tree
Insert the following numbers in the given order:
6, 4, 2, 7, 8, 14, 9

11. Your Turn
Create a binary tree by inserting the following numbers in the given order :
14, 75, 2, 5, 60, 25, 1, 9

12. Your Turn Again
Create a binary tree by inserting the following numbers in the given order :
2, 4, 6, 8, 10
Notice anything weird? It’s unbalanced! 

13. Another Binary Tree h d m a k x

14. Practice w/ Strings
Create a binary tree by inserting the following strings:
Wanda, Alpos, Vazbyte, Fecso, Downs, Mata, Montante

15. Searching in a Binary Tree
Start at the root
Compare against the current node
Found the node if they are equal
Go left if less than current node OR if there is no left, then does not exist
Go right if greater than current node OR if there is no left, then does not exist
Repeat step 2

16. Search(4) – What path do you take?6 2 10 4 1 13

17. Search(9) – What path do you take?6 2 10 4 1 13

18. Removing from a Binary Tree
Search for matching node
If the node is a leaf (0 children), unlink its parent
If the node has 1 child, then link the node’s parent to the node’s child
If the node has 2 children, then go to its right subtree and find the left-most child (smallest value of the right subtree). Take the value and put it in the original node that was being removed. Repeat the remove process on the node with 2 children

19. Remove(4) – 0 children case6 2 10 4 1 13

20. Remove(10) – 1 child case 6 2 10 4 1 13

21. Remove(6) – 2 children case10 4 1 13

22. When do we use binary trees?
...whenever we need to search for quickly
Inherent binary searching capabilities
Large trees can be searched quickly
What is the best case scenario?
What is the worst case scenario?
Balanced tree?
Unbalanced tree?
What is the average case scenario?
Compare a Linked List to a Binary Tree

24. Tree Traversal Add, remove, search only go down 1 path
How do you “walk-through” all the nodes of a tree?

25. In-order tree traversal
Left-subtree traversal if it exists and rerun
Action on the current node (e.g. print)
Right-subtree traversal if it exists and rerun
Note: In-order traversal often used to visit nodes in their inherent order

26. In-order Traversal (1, 2, 4, 6, 8, 10, 13) 6 2 10 4 1 8 13

27. Pre-order tree traversal
Action on the current node (e.g. copy)
Left-subtree traversal if it exists and rerun
Right-subtree traversal if it exists and rerun
Note: Pre-order traversal is often used to duplicate a tree

28. Pre-order Traversal (6, 2, 1, 4, 10, 8, 13)

29. Post-order tree traversal
Left-subtree traversal if it exists and rerun
Right-subtree traversal if it exists and rerun
Action on the current node (e.g. print)
Note: Post-order traversal is often used to completely delete or free up all nodes by visiting children and lowest levels first. (not really necessary with garbage collection)

30. Post-order Traversal (1, 4, 2, 8, 13, 10, 6)

31. Other Resources