1. 1 Trees Definition of a Tree Tree Terminology Importance of Trees Implementation of Trees Binary Trees –Definition –Full and Complete Binary Trees –Implementation and Applications Binary Search Trees –Definition –Importance –Implementation –Insertion –Deletion

2. 2 Definition of a Tree A tree, is a finite set of nodes together with a finite set of directed edges that define parent-child relationships. Each directed edge connects a parent to its child. Example: Nodes={A,B,C,D,E,f,G,H} Edges={(A,B),(A,E),(B,F),(B,G),(B,H), (E,C),(E,D)} A directed path from node m1 to node m k is a list of nodes m 1, m 2,..., m k such that each is the parent of the next node in the list. The length of such a path is k - 1. Example: A, E, C is a directed path of length 2. A B E CD FHG

3. 3 Definition of a Tree (Cont.) A tree satisfies the following properties: 1.It has one designated node, called the root, that has no parent. 2.Every node, except the root, has exactly one parent. 3.A node may have zero or more children. 4.There is a unique directed path from the root to each node. 5 2 416 3 5 2 416 3 5 2 4 1 6 3 treeNot a tree

4. 4 Tree Terminology Ordered tree: A tree in which the children of each node are linearly ordered (usually from left to right). Ancestor of a node v: Any node, including v itself, on the path from the root to the node. Proper ancestor of a node v: Any node, excluding v, on the path from the root to the node. A C B ED F G Ancestors of G proper ancestors of E An Ordered Tree

5. 5 Tree Terminology (Contd.) Descendant of a node v: Any node, including v itself, on any path from the node to a leaf node (i.e., a node with no children). Proper descendant of a node v: Any node, excluding v, on any path from the node to a leaf node. Subtree of a node v: A tree rooted at a child of v. Descendants of a node C A C B ED F G Proper descendants of node B A C B ED F G subtrees of node A

6. 6 Tree Terminology (Contd.) AA B D H C E F G J I proper ancestors of node H proper descendants of node C subtrees of A AA B D H C E F G J I parent of node D child of node D grandfather of nodes I,J grandchildren of node C

7. 7 Tree Terminology (Contd.) Degree: The number of subtrees of a node –Each of node D and B has degree 1. –Each of node A and E has degree 2. –Node C has degree 3. –Each of node F,G,H,I,J has degree 0. Leaf: A node with degree 0. Internal or interior node: a node with degree greater than 0. Siblings: Nodes that have the same parent. Size: The number of nodes in a tree. AA B D H C E F G J I An Ordered Tree with size of 10 Siblings of E Siblings of A

8. 8 Tree Terminology (Contd.) Level (or depth) of a node v: The length of the path from the root to v. Height of a node v: The length of the longest path from v to a leaf node. –The height of a tree is the height of its root node. –By definition the height of an empty tree is -1. AA B D H C E FG J I k Level 0 Level 1 Level 2 Level 3 Level 4 The height of the tree is 4. The height of node C is 3.

9. 9 Importance of Trees Trees are very important data structures in computing. They are suitable for: –Hierarchical structure representation, e.g., File directory. Organizational structure of an institution. Class inheritance tree. –Problem representation, e.g., Expression tree. Decision tree. –Efficient algorithmic solutions, e.g., Search trees. Efficient priority queues via heaps.

10. 10 Implementation of Trees In a general tree, there is no limit to the number of children that a node can have. Representing a general tree by linked lists: –Each node has a linked list of the subtrees of that node. –Each element of the linked list is a subtree of the current node public class GeneralTree extends AbstractContainer { protected Object key ; protected int degree ; protected MyLinkedList list ; //... }

11. 11 Implementation of Trees (Cont.) Definition: An N-ary tree is an ordered tree that is either: 1.Empty, or 2.It consists of a root node and at most N non-empty N-ary subtrees. It follows that the degree of each node in an N-ary tree is at most N. Example of N-ary trees: 5 7 2 59 2-ary (binary) tree B F J DD C GAEB 3-ary (tertiary)tree

12. 12 Implementation of Trees (Cont.) public class NaryTree extends AbstractContainer { protected Object key ; protected int degree ; protected NaryTree[ ] subtree ; public NaryTree(int degree){ key = null ; this.degree = degree ; subtree = null ; } public NaryTree(int degree, Object key){ this.key = key ; this.degree = degree ; subtree = new NaryTree[degree] ; for(int i = 0; i < degree; i++) subtree[i] = new NaryTree(degree); } //... }

13. 13 Binary Trees Definition: A binary tree is an N-ary tree for which N = 2. Thus, a binary tree is either: 1.An empty tree, or 2.A tree consisting of a root node and at most two non-empty binary subtrees. 5 7 2 59 Example:

14. 14 Binary Trees (Contd.) Definition: A full binary tree is either an empty binary tree or a binary tree in which every node is either a leaf node or an internal node with two children. Definition: A complete binary tree is either an empty binary tree or a binary tree in which: 1.Each level k, k > 0, other than the last level contains the maximum number of nodes for that level, that is 2 k. 2.The last level may or may not contain the maximum number of nodes. 3.If a slot with a missing node is encountered when scanning the last level in a left to right direction, then all remaining slots in the level must be empty.

15. 15 Binary Trees (Contd.) Example showing the growth of a complete binary tree:

16. 16 Binary Trees (Contd.) What is the maximum height of a binary tree with n elements? n – 1 What is the minimum height of a binary tree with n elements?  log(n)  What is the minimum and maximum heights of a complete binary tree? Both are  log(n)  What is the minimum and maximum heights of a full binary tree?  log(n)  and  n/2 

17. 17 Binary Trees Implementation rightkeyleft + a* d- bc public class BinaryTree extends AbstractContainer{ protected Object key ; protected BinaryTree left, right ; public BinaryTree(Object key, BinaryTree left, BinaryTree right){ this.key = key ; this.left = left ; this.right = right ; } public BinaryTree( ) { this(null, null, null) ; } public BinaryTree(Object key){ this(key, new BinaryTree( ), new BinaryTree( )); } //... } Example: A binary tree representing a + (b - c) * d

18. 18 Binary Trees Implementation (Contd.) public boolean isEmpty( ){ return key == null ; } public boolean isLeaf( ){ return ! isEmpty( ) && left.isEmpty( ) && right.isEmpty( ) ; } public Object getKey( ){ if(isEmpty( )) throw new InvalidOperationException( ) ; else return key ; } public int getHeight( ){ if(isEmpty( )) return -1 ; else return 1 + Math.max(left.getHeight( ), right.getHeight( )) ; } public void attachKey(Object obj){ if(! isEmpty( )) throw new InvalidOperationException( ) ; else{ key = obj ; left = new BinaryTree( ) ; right = new BinaryTree( ) ; } What is the complexity of each of these operations?

19. 19 Binary Trees Implementation (Contd.) public Object detachKey( ){ if(! isLeaf( )) throw new InvalidOperationException( ) ; else { Object obj = key ; key = null ; left = null ; right = null ; return obj ; } public BinaryTree getLeft( ){ if(isEmpty( )) throw new InvalidOperationException( ) ; else return left ; } public BinaryTree getRight( ){ if(isEmpty( )) throw new InvalidOperationException( ) ; else return right ; } What is the complexity of each of these operations?

20. 20 Applications of Binary Trees Binary trees have many important uses. Two examples are: 1.Binary decision trees. Internal nodes are conditions. Leaf nodes denote decisions. Expression Trees Condition1 Condition2 Condition3 decision1 decision2 decision3 decision4 false True + a* d- bc

21. 21 Binary Search Trees Definition: A binary search tree (BST) is a binary tree that is empty or that satisfies the BST ordering property: 1.The key of each node is greater than each key in the left subtree, if any, of the node. 2.The key of each node is less than each key in the right subtree, if any, of the node. Thus, each key in a BST is unique. Examples: 6 8 2 41 79 53 AA B C D

22. 22 Importance of BSTs DeletionInsertionRetrievalData Structure O(log n) FAST O(log n) FAST O(log n) FAST BST O(n) SLOW O(n) SLOW O(log n) FAST* Sorted Array O(n) SLOW O(n) SLOW O(n) SLOW Sorted Linked List BSTs provide good logarithmic time performance in the best and average cases. Average case complexities of using linear data structures compared to BSTs: *using binary search

23. 23 Implementation of BSTs The BinarySearchTree class inherits the instance variables key, left, and right of the BinaryTree class: public class BinarySearchTree extends BinaryTree implements SearchableContainer { private BinarySearchTree getLeftBST(){ return (BinarySearchTree) getLeft( ) ; } private BinarySearchTree getRightBST( ){ return (BinarySearchTree) getRight( ) ; } //... } What is the complexity of each of these operations?

24. 24 Implementation of BSTs (Cont.) The find method of the BinarySearchTree class: public Comparable find(Comparable comparable) { if(isEmpty()) return null; Comparable key = (Comparable) getKey(); if(comparable.compareTo(key)==0) return key; else if (comparable.compareTo(key)<0) return getLeftBST().find(comparable); else return getRightBST().find(comparable); } What is the complexity of each of find?

25. 25 Implementation of BSTs (Cont.) The findMin method of the BinarySearchTree class: By the BST ordering property, the minimum key is the key of the left-most node that has an empty left-subtree. public Comparable findMin() { if(isEmpty()) return null; if(getLeftBST().isEmpty()) return (Comparable)getKey(); else return getLeftBST().findMin(); } What is the complexity of each of findMin?

26. 26 Implementation of BSTs (Cont.) The findMax method of the BinarySearchTree class: By the BST ordering property, the maximum key is the key of the right-most node that has an empty right-subtree. 20 30 10 154 25 40 35 32 7 9 public Comparable findMax() { if(isEmpty()) return null; if(getRightBST().isEmpty()) return (Comparable)getKey(); else return getRightBST().findMax(); } What is the complexity of each of findMax?

27. 27 Insertion in BSTs By the BST ordering property, a new node is always inserted as a leaf node. The insert method, given in the next page, recursively finds an appropriate empty subtree to insert the new key. It then transforms this empty subtree into a leaf node by invoking the attachKey method: public void attachKey(Object obj) { if(!isEmpty()) throw new InvalidOperationException(); else { key = obj; left = new BinarySearchTree(); right = new BinarySearchTree(); }

28. 28 Insertion in BSTs (Cont.) 1 6 8 2 4 79 3 56 8 2 4 79 3 5 6 8 2 4 79 3 5 6 8 2 4 79 3 5 11 1 public void insert(Comparable comparable){ if(isEmpty()) attachKey(comparable); else { Comparable key = (Comparable) getKey(); if(comparable.compareTo(key)==0) throw new IllegalArgumentException("duplicate key"); else if (comparable.compareTo(key)<0) getLeftBST().insert(comparable); else getRightBST().insert(comparable); } What is the complexity of each of insert?

29. 29 Deletion in BSTs There are three cases: 1.The node to be deleted is a leaf node. 2.The node to be deleted has one non-empty child. 3.The node to be deleted has two non-empty children. CASE 1: DELETING A LEAF NODE Convert the leaf node into an empty tree by using the detachKey method: // In Binary Tree class public Object detachKey( ){ if(! isLeaf( )) throw new InvalidOperationException( ) ; else { Object obj = key ; key = null ; left = null ; right = null ; return obj ; }

30. 30 Deleting a leaf node (cont’d) Example: Delete 5 in the tree below: 7 15 2 41 840 63 9 5 Delete 5 7 15 2 41 840 63 9

31. 31 Deleting a one-child node CASE 2: THE NODE TO BE DELETED HAS ONE NON-EMPTY CHILD (a) The right subtree of the node x to be deleted is empty. Example: 20 35 5 8 3 2240 25 Delete 10 20 35 10 8 5 2240 3 25 6 target temp 6 target // Let target be a reference to the node x. BinarySearchTree temp = target.getLeftBST(); target.key = temp.key; target.left = temp.left; target.right = temp.right; temp = null;

32. 32 Deleting a one-child node (cont’d) (b) The left subtree of the node x to be deleted is empty. Example: Delete 8 7 15 2 41 840 63 12 5 target temp 149 7 15 2 41 1240 63 5 target 14 9 // Let target be a reference to the node x. BinarySearchTree temp = target.getRightBST(); target.key = temp.key; target.left = temp.left; target.right = temp.right; temp = null;

33. 33 CASE 3: DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN DELETION BY COPYING: METHOD#1 Copy the minimum key in the right subtree of x to the node x, then delete the one-child or leaf-node with this minimum key. Example: 7 15 2 41 840 63 9 5 Delete 7 8 15 2 41 940 63 5

34. 34 DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN DELETION BY COPYING: METHOD#2 Copy the maximum key in the left subtree of x to the node x, then delete the one-child or leaf-node with this maximum key. Example: 7 15 2 41 840 63 9 5 Delete 7 6 15 2 41 840 53 9

35. 35 Two-child deletion method#1 code // find the minimum key in the right subtree of the target node Comparable min = target.getRightBST().findMin(); // copy the minimum value to the target target.key = min; // delete the one-child or leaf node having the min target.getRightBST().withdraw(min); All the different cases for deleting a node are handled in the withdraw (Comparable key) method of BinarySearchTree class What is the complexity of each of the delete method?