Ch 10 Trees. Introduction to Trees. Tree Representations. Binary Tree Nodes. Tree Traversals. Binary Search Trees

Transcription

1 Ch 10 Trees Introduction to Trees Tree Representations Binary Tree Nodes Tree Traversals Binary Search Trees 1

2 Binary Trees A binary tree is a finite set of elements called nodes. The set is either empty or consists of a node called the root and two binary trees, the left and right subtrees. The roots of the subtrees are children of the root. An edge connects a node to each of its children. A node is a parent of its children. 2

3 Binary Trees, cont d If n, n,..., n is a list of nodes in a tree such that 1 2 k n is the parent of n for 1 <= i <= k, then this i i+1 list is called a path from n to n. 1 k The length of the path is k 1 (the number of connecting edges.) If there is a path from node R to node M, then R is the ancestor of M, and M is a descendant of R. All nodes in a tree are descendents of the root and the root is the ancester of all nodes. 3

4 Binary Trees, cont d The depth of a node is the length of the path from the root to the node. The height of a tree is one more than the depth of the deepest node. All nodes of depth d are at level d. The root is at level 0 and its depth is 0. A leaf is any node with no non-empty children. An internal node is one with at least one nonempty child. 4

5 Binary Trees, cont d a binary tree is full if All internal nodes have two children AND all leaves are at the same level. In a complete binary tree of height h: All levels except possibly level h 1 are completely full. The bottom level has all of its nodes filled in from the left side. 5

6 Binary Trees, cont d The number of leaves in a non-empty, full binary tree is two more than the number of internal nodes. The number of empty subtrees in a non-empty binary tree is one more than the number of nodes. The number of nodes in a full binary tree of height h is 2 h 1. 6

7 General Trees A general tree is a finite set of nodes. The set is either empty or consists of a root node together with n 0 subtrees. The roots of these subtrees are children of the root which is their parent. All general subtrees have exactly one parent. 7

8 Array Representation Complete binary trees may be implemented with arrays. Position [0] holds the root s data. For a non-leaf node at position [i], the left child is at [2i + 1], the right child is at [2i + 2]. the parent is at [(i - 1)/2] for a non-root node. 8

9 Array Representation A L G O R I T H M S A L G O R I T H M S

10 Linked Representation A L R L L R G L R O R I T L R L R L R L R H M S L R L R L R 10

11 Binary Tree Node: Constructor 11

12 Binary Tree Node: Mutators 12

13 Binary Tree Node: Accessors & Fields 13

14 Binary Trees: Utility Prototypes 14

15 Tree Clear 15

16 Tree Copy 16

17 Tree Size 17

18 Binary Tree Traversals A L G O R I T H M S 18

19 Function Parameters 19

20 Binary Tree Traversals 20

21 Binary Tree Printing A B C D E 21

22 Dichotomous Keys Are you a mammal? Are you bigger than a cat? Do you live underwater? Are you a marsupial? Mouse Trout Robin Kangaroo Raccon 22

23 Dichotomous Keys Are you a mammal? Are you bigger than a cat? Do you live underwater? Are you a marsupial? Mouse Trout Can you fly? Kangaroo Raccon Robin Snake 23

24 Animal Guessing Program 24

25 Animal Guessing Program 25

26 Animal Guessing Program 26

27 Animal Guessing Program 27

28 Binary Search Trees In a binary search tree (BST) the entries of the nodes can be compared with strict weak ordering. Two rules apply to every node: The datum in a node is greater than or equal to all data in its left subtree. The datum in a node is less than all data in its right subtree. 28

29 Inserting into a BST A A R A D V R K AARDVARK 29

30 A Balanced BST H D L B F J N A C E G I K M O 30

31 Removing From a BST if (root is NULL) return else if (target < root) remove target from left subtree else if (target > root) remove target from right subtree else target found, so remove root node 31

32 Removing the Root temp = root pointer if (left subtree is empty) root = right subtree else if (right subtree is empty) root = left subtree else temp = removemax(left subtree) set root s value to temp s value delete temp 32

33 Remove Max if (root s right subtree is not empty) return removemax(right subtree) else temp = root root = left subtree return temp 33

34 The Last Bag 34

35 35