1. Chapter 8 (Lafore’s Book) Binary Tree Hwajung Lee

2.  Definition  Tree is a finite set T which satisfies the following conditions.  It has one root.  Except the root, nodes are divided into n subsets, T 1, T 2, T 3, …, T n. Each subset is also a tree. Thus, tree is defined recursively.

3.  Node: a basic component which can contain data and references to 0 or more other nodes. (ex) A~M  Root: the node on the level 0. (ex) A  Level A B E KL F C G D HIJ M  Level 0  Level 1  Level 2  Level 3

4.  Subtree: a node and all of its descendents (ex) {B, E, F, K, L}, {E, K, L}, …  Leaf (or Terminal node): a node of which degree is 0. (ex) F, G, I, J, K, L, M A B E KL F C G D HIJ M  Level 0  Level 1  Level 2  Level 3

5.  Interior node (or Nonterminal node): a node of which degree is greater than 0.  Child/children  Parent  Siblings: nodes which have the same parent node. (ex) K and L are siblings. H, I and J are siblings. A B E KL F C G D HIJ M  Level 0  Level 1  Level 2  Level 3

6.  Ancestors: the ancestors of a node N are all the nodes on the path from the root to the node N. (ex) the ancestors of node L are A, B, E.  Descendants  Forest: a set of separated trees A B E KL F C G D HIJ M  Level 0  Level 1  Level 2  Level 3

7.  Definition  If every node in a tree can have at most two children, the tree is called a binary tree.  Term  Left child  Right child

8.  Definition  A binary tree in which every node is greater than its left child and less than its right child, if the left and/or right child exists.  class Node class Node  BST operations  find: p378 find  insert (or add): always add new leaf. p380 insert (or add)  traverse  min and max in tree (or subtree)  delete

9.  inorder traversal (in a recursive way)  call itself to traverse the node ’ s left subtree  visit the node  call itself to traverse the node ’ s right subtree

10.  preorder traversl (in a recursive way)  visit the node  call itself to traverse the node ’ s left subtree  call itself to traverse the node ’ s right subtree

11.  postorder traversl (in a recursive way)  call itself to traverse the node ’ s left subtree  call itself to traverse the node ’ s right subtree  visit the node  Example: p386  infix notation, prefix notation, postfix notation

12.  Infix: A*(B+C)  Prefix: *A+BC  Postfix: ABC+*

13.  Deleting a node is the most complicated common operation required for binary search trees.  There are three possible cases when you delete a node.  The node to be deleted is a leaf (had no children). : easy  The node to be deleted has one child. : easy  The node to be deleted has two children. : quite complicated

14.  Case 1: The node to be deleted is a leaf (had no children).  [How?] Change the appropriate child field in the node ’ s parent to point to null.  Java will automatically collect the garbage. Before deletion After deletion

15.  Case 2: The node to be deleted has one child.  [How?] Connect its parent directly to its child. In other words, change the appropriate reference in the parent (leftChild or rightChild) to point to the deleted node ’ s child.

16.  Case 3: The node to be deleted has two children.  [Definition] Inorder successor: For each node, the node with the next- highest key is called its inorder successor, or simply its successor.  [How to delete?] Replace the node to be deleted with its inorder successor and then delete it.  fig 8.16 in page 394  Finding the successor: fig 8.17 in page 395 Finding the successor

17.  Case 3 (continued)  Successor can be the right child of the delNode: fig 8.18 in page 396 ▪ Deletion: fig 8.19 in page 399 ▪ Unplug current from the rightChild field of its parent (or leftChild field if appropriate), and set this filed to point to successor. ▪ Unplug current ’ s left child from current, and plug it into the leftChild field of successor.

18.  Case 3 (continued)  Successor can be a left descendant of the right child of delNode ▪ Deletion: fig 8.20 in page 400 ▪ Plug the right child of successor into the leftChild field of the successor ’ s parent. ▪ Plug the right child of the node to be deleted into the rightChild field of successor. ▪ Unplug current from the rightChild field of its parent, and set this field to point to successor. ▪ Unplug current ’ s left child from current, and plug it into the leftChild field of successor.

19.  Time Complexity of Binary Search Tree: O(log N)O(log N)

20.  the node array[index] ’ s  left child = array[2*index + 1]  right child = array[2*index + 2]  parent = array[(index -1) /2]  fig 8.21 in page 404

21.  Possible – insert() method in the following tree.java will insert a duplicate key as the right child of its twin. However, it is recommended to forbid duplicate keys.

22.  Java code of BST operations Java code of BST operations

23.  Discovered by David Huffman in 1952  The Huffman code uses a binary tree and is applied to data compression. (Ex) [Character Codes] Some ASCII Codes CharacterDecimalBinary ABC…XYZABC…XYZ 65 66 67 … 88 89 90 01000001 01000010 01000011 … 01011001 01011010 01011100

24. Frequency Table (example) CharacterCount A E I S T U Y Space Linefeed 223611241223611241

25. U sp A T If S I Y E 22 9 5 3 2 1 4 2 1 1 13 7 4 2 6 3 2

26. CharacterCode A E I S T U Y Space Linefeed 010 1111 110 10 0110 01111 1110 00 01110 Huffman Code: Textbook (pages 416~420)