Lecture Objectives

1. Lecture Objectives • To learn how to use a tree to represent a hierarchical organization of information • To learn how to use recursion to process trees • To understand the different ways of traversing a tree • To understand the difference between binary trees, binary search trees, and heaps CS340

2. Trees - Introduction 2 • Data organizations • Linear: one predecessor or successor • Nonlinear: multiple predecessors, successors • Accessing all elements in a linear sequence is O(n) • Trees are nonlinear and hierarchical • Tree nodes can have multiple successors (but only one predecessor) CS340

3. Trees - Introduction (cont.) 3 • Trees can represent : • class hierarchy • disk directory and subdirectories • family tree • Trees are recursive data structures • Many methods to process trees are written recursively CS340

4. Binary trees 4 • Each element has two successors • Binary trees can be represented by arrays and linked data structures • Searching in a binary search tree generally more efficient than searching in an ordered list CS340

5. Tree Terminology and Applications CS340

6. Tree Terminology 6 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia science arts mathematics CS340

7. Tree Terminology (cont.) 7 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia The node at the top of a tree is called its root science arts mathematics CS340

8. Tree Terminology (cont.) 8 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia The links from a node to its successors are called branches science arts mathematics CS340

9. Tree Terminology (cont.) 9 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia The successors of a node are called its children science arts mathematics CS340

10. Tree Terminology (cont.) 10 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia Each node in a tree has exactly one parent except for the root node, which has no parent science arts The predecessor of a node is called its parent mathematics CS340

11. Tree Terminology (cont.) 11 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia Nodes that have the same parent are siblings science arts mathematics CS340

12. Tree Terminology (cont.) 12 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia Leaf nodes also are known asexternal nodes, and nonleaf nodes are known as internal nodes A node that has no children is called a leaf node science arts mathematics CS340

13. Tree Terminology (cont.) 13 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia science arts mathematics A generalization of the parent-child relationship is the ancestor-descendant relationship CS340

14. Tree Terminology (cont.) 14 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia science arts mathematics A subtree of a node is a tree whose root is a child of that node CS340

15. Tree Terminology (cont.) 15 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia The level of a node is a measure of its distance from the root plus 1 Level 1 science arts Level 2 mathematics Level 3 CS340

16. Tree Terminology (cont.) 16 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia Level 1 science arts The level of a node is defined recursively Level 2 mathematics Level 3 • If node n is the root of tree T, its level is 1 • If node n is not the root of tree T, its level is 1 + the level of its parent CS340

17. Tree Terminology (cont.) 17 A tree consists of a collection of elements or nodes, with each node linked to its successors wikipedia The height of a tree is the number of nodes in the longest path from the root node to a leaf node science arts The height of this tree is 3 mathematics CS340

18. Binary Trees • In a binary tree, each node has two subtrees • A set of nodes T is a binary tree if either of the following is true • T is empty • Its root node has two subtrees, TL and TR, such that TL and TR are binary trees (TL = left subtree; TR = right subtree) CS340

19. Expression Tree • Each node contains an operator or an operand • Operands are stored in leaf nodes • Tree structure dictates the order of operand evaluation • No parenthesis needed (x + y) * ((a + b) / c) CS340

20. Binary Search Tree • Binary search trees • All elements in the left subtree precede those in the right subtree • A formal definition: A set of nodes T is a binary search tree if either of the following is true: • T is empty • If T is not empty, its root node has two subtrees, TL and TR, such that • TL and TR are binary search trees • the values in the root node of T is greater than all values in TL and is less than all values in TR encyclopedia arts science mathematics

21. Binary Search Tree (cont.) • Does not need to be sorted • Why? • When new elements are inserted (or removed) properly, the binary search tree maintains its order • Compare to an array CS340

22. Binary Search Tree (cont.) • Searching a BST can be O(log n) • What is the worst case? CS340

23. Recursive Algorithm for Searching a Binary Tree • if the tree is empty • return null (target is not found)else if the target matches the root node's data • return the data stored at the root nodeelse if the target is less than the root node's data • return the result of searching the left subtree of the rootelse • return the result of searching the right subtree of the root CS340

24. Full, Perfect, and Complete Binary Trees 7 • Full binary tree: all nodes have either 2 children or 0 children (the leaf nodes) 1 10 2 4 5 6 0 3 9 12 11 13 CS340

25. Full, Perfect, and Complete Binary Trees (cont.) • Perfect binary tree is a full binary tree of height n with exactly 2n – 1 nodes • Find n in this example 3 1 5 0 2 4 6 CS340

26. Full, Perfect, and Complete Binary Trees (cont.) • Complete binary tree is a perfect binary tree through level n - 1 with some extra leaf nodes at level n (the tree height), all toward the left 3 1 5 0 2 4 CS340

27. General Trees • Nodes of a general tree can have any number of subtrees CS340

28. General Trees (cont.) • A general tree can be represented using a binary tree CS340

29. Tree Traversals CS340

30. Tree Traversals • Walking through the tree in a prescribed order and visiting the nodes as they are encountered • Three common kinds of tree traversal • Inorder • Preorder • Postorder CS340

31. Tree Traversals (cont.) • Preorder: visit root node, traverse TL, traverse TR • Inorder: traverse TL, visit root node, traverse TR • Postorder: traverse TL, traverse TR, visit root node CS340

32. Visualizing Tree Traversals • If we keep following the tree to the left, it will trace a route known as the Euler tour CS340

33. Visualizing Tree Traversals (cont.) • A Euler tour (blue path) is a preorder traversal • The sequence in this example isa b d g e h c f i j CS340

34. Visualizing Tree Traversals (cont.) • If we record a node as we return from traversing its left subtree we get an inorder traversal • The sequence isd g b h e a i f j c CS340

35. Visualizing Tree Traversals (cont.) • If we record each node as we last encounter it, we get a postorder traversal • The sequence isg d h e b i j f c a CS340

36. Traversals of Binary Search Trees and Expression Trees • With inorder traversal nodes are visited in sequence Bob, Helen, Peter, Stacy Peter Helen Stacy Bob CS340

37. Traversals of Binary Search Trees and Expression Trees (cont.) * • An inorder traversal of an expression tree results in the sequencex + y * a + b / c • Or with parentheses (x + y) * ((a + b)) / c) + / a b x y + c CS340

38. Traversals of Binary Search Trees and Expression Trees (cont.) * • A postorder traversal x y + a b + c / * • Postfix form of the expression • Operators follow operands + / a b x y + c CS340

39. Traversals of Binary Search Trees and Expression Trees (cont.) * • A preorder traversal * + x y / + a b c • Prefix form of the expression • Operators precede operands + / a b x y + c CS340

40. Implementing a BinaryTreeClass CS340

41. Node<E> Class • The data part is a reference to type E • A binary tree node must have links to both its left and right subtrees CS340

42. Node<E> Class (cont.) protected static class Node<E> implements Serializable { protected E data; protected Node<E> left; protected Node<E> right; public Node(E data) { this.data = data; left = null; right = null; } public String toString() { return data.toString(); } } Node<E> is declared as an inner class within BinaryTree<E> CS340

43. Node<E> Class (cont.) protected static class Node<E> implements Serializable { protected E data; protected Node<E> left; protected Node<E> right; public Node(E data) { this.data = data; left = null; right = null; } public String toString() { return data.toString(); } } Node<E> is declared protected. This way we can use it as a superclass. CS340

44. BinaryTree<E> Class (cont.)

45. BinaryTree<E> Class (cont.) Assuming the tree is referenced by variable bT (type BinaryTree) then . . . CS340

46. BinaryTree<E> Class (cont.) bT.root.data references the Characterobject storing '*' CS340

47. BinaryTree<E> Class (cont.) bT.root.left references the left subtree of the root CS340

48. BinaryTree<E> Class (cont.) bT.root.right references the right subtree of the root CS340

49. BinaryTree<E> Class (cont.) bT.root.right.data references the Characterobject storing '/' CS340

50. BinaryTree<E> Class (cont.)