Chapter 6. Binary Trees. Data Structures and Algorithms in Java
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1 Chapter 6 Binary Trees Data Structures and Algorithms in Java
2 Objectives Discuss the following topics: Trees, Binary Trees, and Binary Search Trees Implementing Binary Trees Searching a Binary Search Tree Tree Traversal Insertion Deletion Data Structures and Algorithms in Java 2
3 Objectives (continued) Discuss the following topics: Balancing a Tree Self-Adjusting Trees Heaps Polish Notation and Expression Trees Case Study: Computing Word Frequencies Data Structures and Algorithms in Java 3
4 Trees, Binary Trees, and Binary Search Trees A tree is a data type that consists of nodes and arcs These trees are depicted upside down with the root at the top and the leaves (terminal nodes) at the bottom The root is a node that has no parent; it can have only child nodes Leaves have no children (their children are null) Data Structures and Algorithms in Java 4
5 Trees, Binary Trees, and Binary Search Trees (continued) Each node has to be reachable from the root through a unique sequence of arcs, called a path The number of arcs in a path is called the length of the path The level of a node is the length of the path from the root to the node plus 1, which is the number of nodes in the path The height of a nonempty tree is the maximum level of a node in the tree Data Structures and Algorithms in Java 5
6 Trees, Binary Trees, and Binary Search Trees (continued) Figure 6-1 Examples of trees Data Structures and Algorithms in Java 6
7 Trees, Binary Trees, and Binary Search Trees (continued) Figure 6-2 Hierarchical structure of a university shown as a tree Data Structures and Algorithms in Java 7
8 Trees, Binary Trees, and Binary Search Trees (continued) An orderly tree is where all elements are stored according to some predetermined criterion of ordering Figure 6-3 Transforming (a) a linked list into (b) a tree Data Structures and Algorithms in Java 8
9 Trees, Binary Trees, and Binary Search Trees (continued) A binary tree is a tree whose nodes have two children (possibly empty), and each child is designated as either a left child or a right child Figure 6-4 Examples of binary trees Data Structures and Algorithms in Java 9
10 Trees, Binary Trees, and Binary Search Trees (continued) In a complete binary tree, all nonterminal nodes have both their children, and all leaves are at the same level A decision tree is a binary tree in which all nodes have either zero or two nonempty children Data Structures and Algorithms in Java 10
11 Trees, Binary Trees, and Binary Search Trees (continued) Figure 6-5 Adding a leaf to tree (a), preserving the relation of the number of leaves to the number of nonterminal nodes (b) Data Structures and Algorithms in Java 11
12 Trees, Binary Trees, and Binary Search Trees (continued) Figure 6-6 Examples of binary search trees Data Structures and Algorithms in Java 12
13 Implementing Binary Trees Binary trees can be implemented in at least two ways: As arrays As linked structures To implement a tree as an array, a node is declared as an object with an information field and two reference fields Data Structures and Algorithms in Java 13
14 Implementing Binary Trees (continued) Figure 6-7 Array representation of the tree in Figure 6.6c Data Structures and Algorithms in Java 14
15 Implementing Binary Trees (continued) Figure 6-8 Implementation of a generic binary search tree Data Structures and Algorithms in Java 15
16 Implementing Binary Trees (continued) Figure 6-8 Implementation of a generic binary search tree (continued) Data Structures and Algorithms in Java 16
17 Implementing Binary Trees (continued) Figure 6-8 Implementation of a generic binary search tree (continued) Data Structures and Algorithms in Java 17
18 Searching a Binary Search Tree Figure 6-9 A function for searching a binary search tree Data Structures and Algorithms in Java 18
19 Searching a Binary Search Tree (continued) The internal path length (IPL) is the sum of all path lengths of all nodes It is calculated by summing Σ(i 1)l i over all levels i, where l i is the number of nodes on level I A depth of a node in the tree is determined by the path length An average depth, called an average path length, is given by the formula IPL/n, which depends on the shape of the tree Data Structures and Algorithms in Java 19
20 Tree Traversal Tree traversal is the process of visiting each node in the tree exactly one time Breadth-first traversal is visiting each node starting from the lowest (or highest) level and moving down (or up) level by level, visiting nodes on each level from left to right (or from right to left) Data Structures and Algorithms in Java 20
21 Breadth-First Traversal Figure 6-10 Top-down, left-to-right, breadth-first traversal implementation Data Structures and Algorithms in Java 21
22 Depth-First Traversal Depth-first traversal proceeds as far as possible to the left (or right), then backs up until the first crossroad, goes one step to the right (or left), and again as far as possible to the left (or right) V Visiting a node L Traversing the left subtree R Traversing the right subtree Data Structures and Algorithms in Java 22
23 Depth-First Traversal (continued) Figure 6-11 Depth-first traversal implementation Data Structures and Algorithms in Java 23
24 Depth-First Traversal (continued) Figure 6-12 Inorder tree traversal Data Structures and Algorithms in Java 24
25 Depth-First Traversal (continued) Figure 6-13 Details of several of the first steps of inorder traversal Data Structures and Algorithms in Java 25
26 Depth-First Traversal (continued) Figure 6-14 Changes in the run-time stack during inorder traversal Data Structures and Algorithms in Java 26
27 Depth-First Traversal (continued) Figure 6-15 A nonrecursive implementation of preorder tree traversal Data Structures and Algorithms in Java 27
28 Stackless Depth-First Traversal Threads are references to the predecessor and successor of the node according to an inorder traversal Trees whose nodes use threads are called threaded trees Data Structures and Algorithms in Java 28
29 Stackless Depth-First Traversal (continued) Figure 6-18 (a) A threaded tree and (b) an inorder traversal s path in a threaded tree with right successors only Data Structures and Algorithms in Java 29
30 Stackless Depth-First Traversal (continued) Figure 6-19 Implementation of the threaded tree and the inorder traversal of a threaded tree Data Structures and Algorithms in Java 30
31 Stackless Depth-First Traversal (continued) Figure 6-19 Implementation of the threaded tree and the inorder traversal of a threaded tree (continued) Data Structures and Algorithms in Java 31
32 Stackless Depth-First Traversal (continued) Figure 6-19 Implementation of the threaded tree and the inorder traversal of a threaded tree (continued) Data Structures and Algorithms in Java 32
33 Traversal Through Tree Transformation Figure 6-20 Implementation of the Morris algorithm for inorder traversal Data Structures and Algorithms in Java 33
34 Traversal Through Tree Transformation (continued) Figure 6-21 Tree traversal with the Morris method Data Structures and Algorithms in Java 34
35 Traversal Through Tree Transformation (continued) Figure 6-21 Tree traversal with the Morris method (continued) Data Structures and Algorithms in Java 35
36 Traversal Through Tree Transformation (continued) Figure 6-21 Tree traversal with the Morris method (continued) Data Structures and Algorithms in Java 36
37 Insertion Figure 6-22 Inserting nodes into binary search trees Data Structures and Algorithms in Java 37
38 Insertion (continued) Figure 6-23 Implementation of the insertion algorithm Data Structures and Algorithms in Java 38
39 Insertion (continued) Figure 6-24 Implementation of the algorithm to insert nodes into a threaded tree Data Structures and Algorithms in Java 39
40 Insertion (continued) Figure 6-24 Implementation of the algorithm to insert nodes into a threaded tree (continued) Data Structures and Algorithms in Java 40
41 Insertion (continued) Figure 6-25 Inserting nodes into a threaded tree Data Structures and Algorithms in Java 41
42 Deletion There are three cases of deleting a node from the binary search tree: The node is a leaf; it has no children The node has one child The node has two children Data Structures and Algorithms in Java 42
43 Deletion (continued) Figure 6-26 Deleting a leaf Figure 6-27 Deleting a node with one child Data Structures and Algorithms in Java 43
44 Deletion by Merging Making one tree out of the two subtrees of the node and then attaching it to the node s parent is called deleting by merging Figure 6-28 Summary of deleting by merging Data Structures and Algorithms in Java 44
45 Deletion by Merging (continued) Figure 6-29 Implementation of algorithm for deleting by merging Data Structures and Algorithms in Java 45
46 Deletion by Merging (continued) Figure 6-29 Implementation of algorithm for deleting by merging (continued) Data Structures and Algorithms in Java 46
47 Deletion by Merging (continued) Figure 6-30 Details of deleting by merging Data Structures and Algorithms in Java 47
48 Deletion by Merging (continued) Figure 6-31 The height of a tree can be (a) extended or (b) reduced after deleting by merging Data Structures and Algorithms in Java 48
49 Deletion by Merging (continued) Figure 6-31 The height of a tree can be (a) extended or (b) reduced after deleting by merging (continued) Data Structures and Algorithms in Java 49
50 Deletion by Copying If the node has two children, the problem can be reduced to: The node is a leaf The node has only one nonempty child Solution: replace the key being deleted with its immediate predecessor (or successor) A key s predecessor is the key in the rightmost node in the left subtree Data Structures and Algorithms in Java 50
51 Deletion by Copying (continued) Figure 6-32 Implementation of an algorithm for deleting by copying Data Structures and Algorithms in Java 51
52 Deletion by Copying Figure 6-32 Implementation of an algorithm for deleting by copying (continued) Data Structures and Algorithms in Java 52
53 Deletion by Copying Figure 6-32 Implementation of an algorithm for deleting by copying (continued) Data Structures and Algorithms in Java 53
54 Deletion by Copying (continued) Figure 6-33 Deleting by copying Data Structures and Algorithms in Java 54
55 Balancing a Tree A binary tree is height-balanced or balanced if the difference in height of both subtrees of any node in the tree is either zero or one A tree is considered perfectly balanced if it is balanced and all leaves are to be found on one level or two levels Data Structures and Algorithms in Java 55
56 Balancing a Tree (continued) Figure 6-34 Different binary search trees with the same information Data Structures and Algorithms in Java 56
57 Balancing a Tree (continued) Figure 6-35 Maximum number of nodes in binary trees of different heights Data Structures and Algorithms in Java 57
58 Balancing a Tree (continued) Figure 6-36 Creating a binary search tree from an ordered array Data Structures and Algorithms in Java 58
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