topic: trees

1. Data Structures and Algorithms
Trees - II

2. Outline
• Binary Tree Revision
• Traversing a Binary Tree
– Depth-first
• Pre-order
• In-order
• Post-order
– Breadth-first

3. Tree Traversal
• A traversal iterates through a collection, one item at a time, in
order to access or visit each item.
• With a linear structure such as a linked list, the traversal is
rather easy since we can start with the first node and iterate
through the nodes, one at a time, by following the links
between the nodes.
• But how do we visit every node in a binary tree?

4. Linked Implementation

5. Pre-order Traversal
Since every node is the
root of its own subtree,
we can repeat the same
process on each node,
resulting in a
recursive solution.
(Also called NLR traversal)

6. In-order Traversal
(Also called LNR traversal)

7. Post-Order Traversal
(Also called LRN traversal)

8. Depth-First and Breadth-First Traversal
• The preorder, inorder, and postorder traversals are
all examples of a depth-first traversal.
– That is, the nodes are traversed deeper in the tree
before returning to higher-level nodes.
• Another type of traversal that can be performed on
a binary tree is the level order traversal which is
known as breadth-first traversal.
– That is, the nodes are visited by level, from left to right.

9. Level-Order Traversal

10. Expression Trees
• Arithmetic expressions such as (9+3)*(8 4) can be
represented using an expression tree.
• An expression tree is a binary tree in which the
operators are stored in the interior nodes and the
operands (the variables or constant values) are
stored in the leaves.
• Once constructed, an expression tree can be used
to evaluate the expression or for converting an
infix expression to either pre-fix or post-fix
notation.

11. Expression Tree (Cont.)
• The structure of the expression tree is based on the
order in which the operators are evaluated.
• The operator in each internal node is evaluated after
both its left and right subtrees have been evaluated.
• The root node contains the operator to be evaluated.

12. +*ab/cd
Exercise
• Write the expression obtained after traversing
the following tree via Pre-order Traversal

13. a+b+c+d
Exercise
• Write the expression obtained after traversing
the following tree via in-order Traversal

14. ab+cd*/
Exercise
• Write the expression obtained after traversing
the following tree via post-order Traversal
/
+
d
c
b
a
*

15. *
+
9
/
6
5
Practice Question
Write the Preorder, Inorder and
Postorder traversals of the given
binary tree.
Also write the level order
traversal.
-
2
4

16. References
• Data Structures and Algorithms Using Python, Rance D.
Necaise, John Wiley & Sons, Inc. 2010
• John R. Hubbard, Data Structures with Java (Schaum's Outline
Series), McGraw-Hill Education; 2nd edition (June 16, 2009)