1. Binary Tree

2. Some Terminologies Short review on binary tree Tree traversals Binary Search Tree (BST)‏ Questions

3. Some Terminologies Child and Parent  Every node except the root has one parent.  A node can have zero or more children. Leaves  Leaves are nodes with no children.  1,3,7,8 Sibling  nodes with the same parent.  2 and 8; 1 and 4; 3 and 7

4. Some Terminologies Path  A sequence of edges.  Length of a path: number of edges on the path Depth of a node = no. of edges from root  Root is also a node, depth of root = ? Height of a node = length of longest path to a leaf  Height of leaves = ?  Height of 2 = ?  Height of root = height of tree

5. Binary Tree Some Terminologies Short review on binary tree Tree traversals Binary Search Tree (BST)‏ Questions

6. Short review on binary tree At most two children for each node A root node at the top (or the tree is empty)‏ Nodes with no child=leaf nodes Left complete tree if:  All levels are full except last level  Last level filled from left to right

7. Binary Tree Some Terminologies Short review on binary tree Tree traversals Binary Search Tree (BST)‏ Questions

8. Tree traversals To visit each node of the tree Recursively visiting left and right sub-trees Pre-order (NLR)‏ Node  Current Node, Left sub-tree, Right sub-tree In-order (LNR)‏ Node  Left, Node, Right Post-order (LRN)‏ Node  Left, Right, Node

9. Tree traversals Examples Pre-order (NLR)‏  9,4,2,3,8,6,7  Always print the node first 9 4 23 8 67

10. Tree traversals Examples In-order (LNR)‏  2,4,3,9,6,8,7  Print whenever return from left 9 48 2367 48 9

11. Tree traversals Examples Post-order (LRN)‏  2,3,4,6,7,8,9  Print whenever return from right 9 48 2367 4 9 8

12. Binary Tree Some Terminologies Short review on binary tree Tree traversals Binary Search Tree (BST)‏ Questions

13. Binary Search Tree Properties A Binary Tree (nodes at most two children)‏ Node Left sub-tree < Node < Right sub-tree Which one(s) is/are BST(s)? (1)‏(2)‏ (3)‏

14. Binary Search Tree Which one(s) is/are BST(s)? (2)‏ (1)‏ (3)‏

15. Binary Search Tree Which one(s) is/are BST(s)? (1)‏(2)‏ (3)‏

16. Binary Search Tree Summary LeftNodeRight BST: Left < Node < Right  Heap: Only requiring Parent > Children WHOLEsub-tree WHOLE left (or right) sub-tree X)‏  All items in the sub-tree has to be X)‏ NOT necessarily a Complete tree! Worse case can be a linked list! Same collection of elements can have different BSTs

17. Binary search tree Find key To look for a node with a certain Key Left < Node < Right Recursively do: 1. Compare(key,node)‏ 2. Key<node: Go left 3. Key>node: Go right

18. Binary search tree Find the key 8 node 6, Key > 6 Go right, node 8, Key == 8 >

19. Binary search tree Find the key 3 node 6, Key < 6 Go left, node 2, Key > 2 Go right, node 4, Key < 4 Go left, node 3, Key == 3 O(1) at each level Find a key  O(depth) if found  O(height) if not found Time complexity=O(height)‏ < > <

20. Binary search tree FindMin and FindMax Left < Node < Right Min = Left most node  Recursively go left Until no more left child Max = Right most node  Recursively go right Until no more right child

21. Binary search tree FindMin and FindMax

22. Binary search tree Insert To insert a key into BST, similar to Find Key Proceed as if you want to find the key If found, duplicate key do nothing; or update a counter; or insert to a list; or … Otherwise insert X at the last spot.

23. Binary search tree Insert the key 5 node 6, Key < 6 Go left, node 2, Key > 2 Go right, node 4, Key > 4 Go right, NULL, insert 5 O(1) at each level Again, O(height)‏ < > 5

24. Binary search tree Delete To delete a key from a BST Proceed as if you want to find the key If found the node n, delete it! that simple Is it really that simple?  I wish it was  Step two is a bit harder: three cases

25. Binary search tree Delete the key 1 node 6, Key < 6 Go left, node 2, Key < 2 Go left, node 1, Key == 1 Case one: no child Delete it! < <

26. Binary search tree Delete the key 8 node 6, Key > 6 Go right, node 8, Key == 8 Case two: one child My parent bypass me, point to my child instead Really no violation?  Left < X < Right >

27. Binary search tree Delete the key 2 node 6, Key < 6 Go left, node 2, Key == 2 Case two: two child Replace node 2 with Min node (m) of right sub-tree Recursively delete m Why choose this way? <

28. Binary search tree Delete the key 2, before and after

29. Binary search tree Delete Summary Case 1: node n is a leaf (no child): delete it! Case 2: node n has only one child:  Parent bypass n, point to n.child and delete n Case 3: node n has two child:  Replace n with smallest node (m) of the right sub-tree  Recursively delete m O(height)? Smallest node=left-most node  Either has right child; OR No child  So, delete m can only be case 1 or 2, the simple cases