CS24 Week 8 Lecture 1

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1 CS24 Week 8 Lecture 1 Kyle Dewey

2 Overview Tree terminology Tree traversals Implementation (if time)

3 Terminology

4 Node The most basic component of a tree - the squares

5 Edge The connections between nodes - the arrows

6 Parent / Child A parent is the predecessor of a node A child is the successor of a node Not all nodes have parents Not all nodes have children (parent of 3 and 12) (child of ) (child of )

7 Leaf / Terminal Node A node without any children

8 Leaf / Terminal Node A node without any children (leaf) (leaf)

9 Internal Node A node with at least one child

10 Internal Node A node with at least one child (internal node)

11 Root Node Node without any parent Often drawn as the topmost node

12 Height and Depth Height: The number of edges on the longest path from a node to a leaf Depth: the number of edges between a node and the root node

13 Height and Depth

14 Level All the nodes of a tree which have the same depth

15 Level All the nodes of a tree which have the same depth level 0 level 1 level 2

16 k-ary Tree A tree where each node can have between 0 and k children What is k for a binary tree?

17 k-ary Tree A tree where each node can have between 0 and k children What is k for a binary tree? - 2

18 Full k-ary Tree All nodes have either 0 or k children

19 Complete k-ary Tree Like a full k-ary tree, except the last level is permitted to be missing nodes, but only on the right 0 4

20 Complete k-ary Tree Like a full k-ary tree, except the last level is permitted to be missing nodes, but only on the right ok 0 4

21 Complete k-ary Tree Like a full k-ary tree, except the last level is permitted to be missing nodes, but only on the right 0 10

22 Complete k-ary Tree Like a full k-ary tree, except the last level is permitted to be missing nodes, but only on the right not ok 0 10

23 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one

24 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one ok

25 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one 0 15

26 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one ok 0 15

27 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one

28 Balanced Tree For all nodes, the height of the left and right subtrees differ by no more than one not ok

29 Subtree Nearly synonymous with node We recursively defined the tree to be either a node with an element and two children, or an empty tree (NULL) Generally refers to some subcomponent of a larger tree, including recursive subcomponents

30 Subtree Example Can refer to 3 and its children Cannot just refer to 3

31 Traversals

32 Traversals For many tree-related problems, the order in which nodes are processed can have a huge impact Two basic kinds: breadth-first search and depth-first search

33 Breath-First Search (BFS) Tree is traversed as if nodes were words on a page (top to bottom, left to right)

34 Breath-First Search (BFS) Tree is traversed as if nodes were words on a page (top to bottom, left to right)

35 Implementing BFS Question: how might we implement BFS? Hint: you ll need a data structure you ve implemented before

36 Implementing BFS Idea: put nodes on a queue Visit nodes according to the queue order When we are done with a node, put its children onto the end of the queue

37 Implementing BFS root Queue: <<empty>>

38 Implementing BFS root Put root on the queue first (this is the node, not just the Queue: number)

39 Implementing BFS root Now dequeue Queue:

40 Implementing BFS root 1 Now dequeue Queue:

41 Implementing BFS root 1 Now dequeue Queue: <<empty>>

42 Implementing BFS root 1 Now put on the child nodes Queue: <<empty>>

43 Implementing BFS root 1 Now put on the child nodes Queue: 3, 12

44 Implementing BFS root 1 Repeat Queue: 3, 12

45 Implementing BFS root 1 2 Queue: 3, 12

46 Implementing BFS root 1 2 Queue: 12

47 Implementing BFS root 1 2 Queue: 12, 0, 4

48 Implementing BFS root 1 2 Queue: 12, 0, 4

49 Implementing BFS root Queue: 12, 0, 4

50 Implementing BFS root Queue: 0, 4

51 Implementing BFS root Queue: 0, 4, 10, 15

52 Implementing BFS root Queue: 0, 4, 10, 15

53 Implementing BFS root Queue: 0, 4, 10, 15

54 Implementing BFS root Queue: 4, 10, 15

55 Implementing BFS root Queue: 4, 10, 15

56 Implementing BFS root Queue: 10, 15

57 Implementing BFS root Queue: 10, 15

58 Implementing BFS root Queue: 15

59 Implementing BFS root Queue: 15

60 Implementing BFS root Queue: <<empty>>

61 Depth-First Search (DFS) Favor going down towards the left first

62 Implementing DFS Question: how might we implement DFS? Hint: you ll need a data structure you ve implemented before

63 Implementing DFS Idea: put nodes on a stack Visit nodes according to the stack order When we are done with a node, push its children onto the top of the stack

64 Implementing DFS root Stack: <<empty>>

65 Implementing DFS root Stack:

66 Implementing DFS root Stack:

67 Implementing DFS root 1 Stack:

68 Implementing DFS root 1 Stack: <<empty>>

69 Implementing DFS root 1 Stack: 3, 12

70 Implementing DFS root 1 Stack: 3, 12

71 Implementing DFS root 1 2 Stack: 3, 12

72 Implementing DFS root 1 2 Stack: 12

73 Implementing DFS root 1 2 Stack: 0, 4, 12

74 Implementing DFS root 1 2 Stack: 0, 4, 12

75 Implementing DFS root Stack: 0, 4, 12

76 Implementing DFS root Stack: 4, 12

77 Implementing DFS root Stack: 4, 12

78 Implementing DFS root Stack: 12

79 Implementing DFS root Stack: 12

80 Implementing DFS root Stack: <<empty>>

81 Implementing DFS root Stack: 10, 15

82 Implementing DFS root Stack: 10, 15

83 Implementing DFS root Stack: 10, 15

84 Implementing DFS root Stack: 15

85 Implementing DFS root Stack: 15

86 Implementing DFS root Stack: <<empty>>

87 On Using Stacks We can cut out the explicit stack by using the call stack implicitly via recursion void traverse(node* current) { if (current!= NULL) { traverse(current->getleft()); traverse(current->getright()); } }

88 Specific Kinds of DFS Traversals Depending on when we process the current node, there are three general kinds of DFS traversals: Pre-order: process current first In-order: process current between left and right Post-order: process current after left and right

89 Pre-Order Traversal void traverse(node* current) { if (current!= NULL) { process(current); traverse(current->getleft()); traverse(current->getright()); } }

90 In-Order Traversal void traverse(node* current) { if (current!= NULL) { traverse(current->getleft()); process(current); traverse(current->getright()); } }

91 Post-Order Traversal void traverse(node* current) { if (current!= NULL) { traverse(current->getleft()); traverse(current->getright()); process(current); } }

92 Using Traversals Say we want to print out the contents of a binary search tree in sorted order What kind of traversal should we use?

93 Using Traversals Say we want to print out the contents of a binary search tree in sorted order What kind of traversal should we use? - inorder

94 Using Traversals Say we want to delete a binary search tree Which traversal is best?

95 Using Traversals Say we want to delete a binary search tree Which traversal is best? - post-order

Topic 18 Binary Trees "A tree may grow a thousand feet tall, but its leaves will return to its roots." -Chinese Proverb

Topic 18 Binary Trees A tree may grow a thousand feet tall, but its leaves will return to its roots. -Chinese Proverb Topic 18 "A tree may grow a thousand feet tall, but its leaves will return to its roots." -Chinese Proverb Definitions A tree is an abstract data type one entry point, the root Each node is either a leaf