1. Trees
Chapter 9

2. Tree graph connected undirected no simple circuits (acyclic)
no multiple edges
no loops

3. Sample Trees?
Tree Tree Not Not

4. Theorem 1 An undirected graph is a tree iff
A simple path exists in a tree
between any two vertices

5. Root A particular tree vertex Each edge is directed away from the root
from which we assign a direction to each edge
Each edge is directed away from the root

6. Rooted tree A tree with a designated root A directed graph
Direction of all edges is away from root

7. Parent In a rooted tree, a parent of vertex v is the unique vertex u
such that there is a directed edge from u to v

8. Child The vertex v to which a directed edge exists
from parent u in a rooted tree

9. Siblings
vertices with the same parent

10. Leaf
a vertex of a tree that has no children

11. Ancestors of node A
nodes located on the path
from A to the root

12. Descendants of node A
nodes located on the path
from A to a leaf node

13. Internal vertices
Vertices with children

14. Sub-tree a tree contained in a larger tree
whose root may be a child node
in the larger tree

15. m-ary tree
a rooted tree
with no more than m children per vertex

16. Full m-ary tree a rooted tree whose every internal vertex
has exactly m children

17. Theorem 2
A tree with n vertices has n - 1 edges.
7 vertices
6 edges

18. Theorem 3 A full m-ary tree with i internal vertices
contains n = mi + 1 vertices.
m = 2
i = 7
15 vertices

19. Tree Height height (level) of a node height of a tree
the length of the path from the root to a node
height of a tree
the length of the longest path in a tree

20. The maximum number of nodes at any level is mh
h is height of a node at that level of the tree
212223

21. The minimum number of nodes
of a tree of height h is
h+1

22. The maximum number of nodes
in a tree of height h is
m(h+1) -1
2(3+1) - 1

23. Balanced tree A rooted m-ary tree of height h is called balanced
if all leaves are at level h or h - 1
YES NO
YES

24. If an m-ary tree of height h has l leaves,
and the tree is full and balanced,
h = ceil(log m l)
h = ceil (log28)
h = 3
What does this imply about access speed if a tree is used as a data structure?

25. Applications of Trees
8.2

26. Binary search tree A binary tree where key value in any node is
greater than key of its left child
and any of its children
(the nodes in the left subtree)
less than key of its right child
(the nodes in the right subtree)

27. Binary Search Tree Example

28. Form a BST with the words Mathematics, Physics, Geography, Zoology, Meteorology, Geology, Psychology, Chemistry

29. NOTE: Input order determines a tree's shape. Tree Animation

30. Tree Traversal
8.3

31. Inorder Tree Traversal
process Left subtree inorder
Visit a node (or process node)
Process Right subtree inorder
Processes BST vertices in ascending sequence

32. Inorder Traversal Example LVR
Arps, Dietz, Egofske, Fairchild, Garth, Huston, Keith
Magillicuddy, Nathan, Perkins, Seliger, Talbot,
Underwood,Verkins, Zarda

33. Preorder Tree Traversal
allows quickest access to the whole tree
VISIT a node
process LEFT subtree in preorder
process RIGHT subtree in preorder

34. Preorder Traversal Example VLR
Magillicuddy, Fairchild, Dietz, Arps, Egofske, Huston,
Garth, Keith, Talbot, Perkins, Nathan, Selinger, Verkins,
Underwood, Zarda

35. Postorder Tree Traversal
good for deletion of nodes; postfix notation
process LEFT subtree in postorder
process RIGHT subtree in postorder
VISIT a node

36. Postorder Traversal Example LRV
Arps, Egofske, Dietz, Garth, Keith, Huston, Fairchild,
Nathan, Selinger, Perkinds, Underwood, Zarda, Verkins,
Talbot, Magillicuddy

37. Expression Tree An ordered rooted tree
associates operands & operators in a uniform way +

38. Give Pre, In, Postorder PreOrder: InOrder: PostOrder:
- + + * * 8 3 / 6 7
InOrder:
6 * * / 7
PostOrder:
6 2 * * / - +

39. Spanning Trees
9.4

40. Spanning Subgraph A spanning subgraph of G is G’ = (V, E’)
where E’ is a subset of E
Note every vertex of G is included

41. Spanning Tree A spanning subgraph that is a tree connected acyclic
See p

42. Depth First Search A procedure for constructing a spanning tree
by adding edges that form a path until this is not possible
then moving back up the tree
until a vertex is found where a new path can be formed

43. Breadth First Search A procedure for constructing a spanning tree
that successively adds all edges incident to the last set of edges added
unless a simple circuit is formed

44. Perform DFS, BFS search
DFS: a, b, c, d, e, f, g BFS: a b c g d e f

45. Perform DFS, BFS search
DFS: a, b, c, d, e, f BFS: a b d e c f

46. Minimum Spanning Tree A connected weighted graph
is a spanning tree that has
the smallest possible sum of weights of its edges.