birds fly S NP N VP V Graphs and trees

Transcription

1

2 birds fly S NP VP N birds V fly S NP NP N VP V VP

3 S NP VP birds a fly b ab = string

4 S A B a ab b S A B A a B b

5 S NP VP birds a fly b ab = string

6 Grammar 1: Grammar 2: A a A a A a B A B a B b A B A b Grammar 3: Grammar 4: A a A a A a B A B a B b B b B b A B A b

7 Grammar 5: Grammar 6: S a A A a S b B A B a A a S B b B b b S B b A S Grammar 7: Grammar 8: A a A a A a A a A a b A b A b

8 Trees consist of a set of nodes

9

10 NODE NODE

11 NODES NODES

12 ROOT NODE ROOT NODE NODES NODES

13 Trees with a distinct root node are called rooted trees

14 a f b d e c g NON-ROOTED TREE

15 f d e b g a c

16 d b e a c f g ROOTED TREE

17 The connnections between the nodes are called edges

18 EDGES EDGE

19 EDGES NODES NODES EDGE

20 A rooted tree is a collection of nodes, one of which is distinguished as a root, along with a relation (parenthood) that places a hierarchical structure on the nodes. (Avo, Hopcroft, Ullman 1983: 75)

21 The connections between nodes are determined by specific rules.

22 Some properties of rooted trees

23 Property 1: A rooted tree has only one root node

24 ROOT ROOTS a tree not a tree

25 Property 2: Except for the root node, all the nodes in a rooted tree have exactly one parent (mother)

26 ROOT NODE PARENT NODE ROOT NODE PARENT NODES

27 DISALLOWED: ROOT NODE PARENT NODES

28 Property 3: Nodes can have any number of children (daughter) nodes

29 PARENT NODE 2 CHILDREN NODES 1 CHILD PARENT

30 NODES ROOT/PARENT DAUGHTER/ PARENT DAUGHTER

31 How many child nodes are there?

32 How many parent and child nodes are there?

33 Paths

34 A path is a sequence of nodes

35 PATHS

36 The length of a path is one less than the number of nodes in the path

37 PATHS

38 In a path from a node a to a node b, with node a higher in the tree than node b (see below for node height), node a is an ancestor of node b and node b is a descendant of node a

39 E.g. 4, 5, 10, 11, 12 are descendants of 2 1 and 2 are ancestors of 4 and 5

40 Leaves

41 A leaf is a node that does not have descendants. (Avo, Hopcroft, Ullman 1983: 76)

42 A B C D E F G H I J K L M LEAVES

43 Height

44 The height of node in a tree is the length of the longest path to a leaf

45 A B C D E F G H I 2 J K L M The height of C is 2

46 The height of a tree is the height of the root (ibid.)

47 A B C D E F G H I 3 J K L M The height of the tree is 3

48 Some properties of trees For any tree, the following three statements hold:

49 Properties of trees trees

50 1. Starting from any node, any other node in the tree can be reached. 2. Nodes have an arbitrary number of children (daughter) nodes. 3. The number of edges n-1 in a tree is always one less than the number of nodes n.

51 1. Starting from any node, any other node in the tree can be reached. Therefore, there is no node that cannot be reached through some simple path.

52 Trace all possible paths to all possible nodes

53 B NO PATH TO NODE B

54 2. There are no cycles. A cycle exists when, starting from some node v, there is some path that travels through some set of nodes v1, v2,..., vk that then arrives back at v.

55 A TREE v v1 v2 NOT A TREE v v1 v2

56 3. The number of edges in a tree is always one less than the number of nodes

57 edges = 2, nodes = 3 edges = 1, nodes = 2 How many edges? How many nodes?

58 1. Starting from any node, any other node in the tree can be reached. There exists no node that cannot be reached through some simple path. 2. There are no cycles. A cycle exists when, starting from some node v, there is some path that travels through some set of nodes v1, v2,..., vk that then arrives back at v. 3. The number of edges n-1 in a tree is always one less than the number of nodes n.

59 a f b d e c g TREE

60 Representation Trees can also be represented as a list or a nest of brackets

61 TREE A b c LIST A b c BRACKETS (A (b c))

62 TREE A B LIST A B c d c d BRACKETS (A ( B (c d)))