This document discusses trees and tree traversal algorithms. It defines trees and forests, and introduces spanning trees and spanning forests. It then discusses ordered rooted trees and the three tree traversal algorithms: preorder, inorder, and postorder. Preorder traverses the root first, then left subtree, then right subtree. Inorder traverses left subtree, then root, then right subtree. Postorder traverses left subtree, then right subtree, then root. Examples of each traversal type on sample trees are provided.

1. Discrete Structures
Trees

2. Trees and Forests
• A (free) tree is an undirected graph T
such that
– T is connected
– T has no cycles (circuits)
This definition of tree is different
from the one of a rooted tree
• A rooted tree is a tree in which one
vertex has been designated as the
root and every edge is directed away
from the root.
• A forest is an undirected graph
without cycles
• The connected components of a
forest are trees
Tree
Forest

3. Spanning Trees and Forests
• A spanning tree of a connected
graph is a spanning subgraph that
is a tree
• A spanning tree is not unique
unless the graph is a tree
• Spanning trees have applications
to the design of communication
networks
• A spanning forest of a graph is a
spanning subgraph that is a forest
Graph
Spanning tree

4. Tree Traversal
• Ordered Rooted Tree:
An ordered rooted tree is a rooted tree where
the children of each internal vertex are
ordered.
• Tree Traversal:
Ordered rooted trees are often used to store
information. Tree traversal is the procedure of
visiting different vertices of the tree to read
information stored in that vertex. There are
three different orders of the tree traversal.

5. Inorder Traversal
• Let T be an ordered rooted tree with root r
and T1, T2, …Tn are the subtrees at r from left to
right. The in order traversal begins by
traversing T1 in inorder, then visiting r. It
continues by traversing T2 in inorder, then T3 in
inorder, . . . , and finally Tn in inorder.

6. Inorder Tree Walk
• Left, root, right
• Example:
• A, B, D, F, H, K
F
B H
KDA

7. Inorder Tree Walk
• What does the following code do?
TreeWalk(x)
TreeWalk(left[x]);
print(x);
TreeWalk(right[x]);
• A: prints elements in sorted (increasing) order
• This is called an inorder tree walk
– Preorder tree walk: print root, then left, then right
– Postorder tree walk: print left, then right, then root

8. Example – Inorder Traversal
• Example 3, p-714

9. Preorder Traversal
• Let T be an ordered rooted tree with root r
and T1, T2, …Tn are the subtrees at r from left to
right. The preorder traversal begins by visiting
r . It continues by traversing T1 in preorder,
then T2 in preorder, then T3 in preorder, . . . ,
and finally Tn in preorder.

10. Preorder Traversal
• Root, left, right
• Example:
• F, B, A, D, H, K
F
B H
KDA

11. Preorder Traversal
• Example 2, p-714

12. Postorder Traversal
• Let T be an ordered rooted tree with root r
and T1, T2, …Tn are the subtrees at r from left to
right. The preorder traversal begins by
traversing T1 in postorder, then T2 in postorder,
then T3 in postorder, . . . , and finally Tn in
postorder and ends by visiting r.

13. Postorder Traversal
• Left, right, root
• Example:
• A, D, B, K, H, F
F
B H
KDA

14. Postorder Traversal
• Example 4, p-714
02/09/15

15. A Shortcut for Traversing an Ordered
Rooted Tree in Preorder, Inorder, and Postorder.
• See Q. 7 to 15, p-723