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Lecture Objectives

This lecture explores the fundamental concepts of tree data structures, highlighting their role in representing hierarchical organizations of information. Topics include the use of recursion in tree processing, various tree traversal methods, and differences among binary trees, binary search trees, and heaps. Learn how trees serve as recursive data structures with applications in class hierarchies, family trees, and directory structures. By the end, students will understand key terminology, the relationship between nodes, and the significance of tree height in computer science applications.

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Lecture Objectives

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  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.)

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