1. Week 7 - Friday

2.  What did we talk about last time?  Trees in general  Binary search trees

4. Infix to Postfix Converter

7. public class Tree { private static class Node { public int key; public Object value; public Node left; public Node right; } private Node root = null; … } The book uses a generic approach, with keys of type Key and values of type Value. The algorithms we'll use are the same, but I use int keys to simplify comparison.

8.  Almost all the methods we call on trees will be recursive  Each will take a Node reference to the root of the current subtree  Because the root is private, assume that every recursive method is called by a public, non-recursive proxy method: public Type doSomething() calls private static Type doSomething( Node root )

9. private static Node add(Node node, int key, Object value) Proxy: public void add(int key, Object value) { root = add( root, key, value ); } Find where the node should be. If there is already a matching key, do nothing. Otherwise, put a new node at the null location. Note: This can cause an unbalanced tree.

11.  Let h be the height of a binary tree  A perfect binary tree has 2 h+1 – 1 nodes  Alternatively, a perfect binary tree has 2L – 1 nodes, where L is the number of leaves  A complete binary tree has between 2 h and 2 h+1 – 1 (exclusive) nodes  A binary tree with n nodes has n + 1 null links

12.  Unbalanced BST:  O(n) find  O(n) insert  O(n) delete  Balanced BST:  O(log n) find  O(log n) insert  O(log n) delete

14.  Visiting every node in a tree is called a traversal  There are three traversals that we are interested in today:  Preorder  Postorder  Inorder  We'll get to level order traversal in the future

15.  Preorder:  Process the node, then recursively process its left subtree, finally recursively process its right subtree  NLR  Postorder:  Recursively process the left subtree, recursively process the right subtree, and finally process the node  LRN  Inorder:  Recursively process the left subtree, process the node, and finally recursively process the right subtree  LNR

16. 4 4 2 2 5 5 1 1 3 3 6 6 4 2 1.. 3.. 5. 6..

17. .. 1.. 3 2... 6 5 4 4 4 2 2 5 5 1 1 3 3 6 6

18. . 1. 2. 3. 4. 5. 6. 4 4 2 2 5 5 1 1 3 3 6 6

19. 10 6 6 14 1 1 9 9 17 7 7 2 2 15

21. public class Tree { private static class Node { public int key; public Object value; public Node left; public Node right; } private Node root = null; … } The book uses a generic approach, with keys of type Key and values of type Value. The algorithms we'll use are the same, but I use int keys to simplify comparison.

22. private static void preorder( Node node ) Proxy: public void preorder() { preorder( root ); } Just print out each node (or a dot). Real traversals will actually do something at each node.

23. private static void inorder( Node node ) Proxy: public void inorder() { inorder( root ); } Just print out each node (or a dot). Real traversals will actually do something at each node.

24. private static void postorder( Node node ) Proxy: public void postorder() { postorder( root ); } Just print out each node (or a dot). Real traversals will actually do something at each node.

25.  We can take the idea of an inorder traversal and use it to store a range of values into a queue  We want to store all values greater than or equal to the min and less than the max private static void getRange( Node node, Queue queue, int min, int max ) Proxy: public Queue getRange(int min, int max){ Queue queue = new ArrayDeque (); getRange( root, queue, min, max ); return queue; }

28. private static Node delete(Node node, int key) Proxy: public void delete(int key) { root = delete( root, key ); } 1. Find the node 2. Find its replacement (smallest right child or largest left child) 3. Swap out the replacement We may need some subroutines: private static Node smallest(Node node, Node parent) private static Node largest(Node node, Node parent) Note: This can cause an unbalanced tree.

30.  Finish delete  Breadth first traversal  Balancing binary search trees  2-3 search trees  Red-black trees

31.  Finish Project 2  Due tonight by midnight  Read section 3.3  Start Assignment 4  Due next Friday  Office hours from 3:30-5pm canceled today due to travel