1. Transforming Infix to Postfix
Steps to convert the infix expression a / b * ( c + ( d – e ) ) to postfix form.

2. Infix-to-Postfix Algorithm
Symbol in Infix
Action
Operand
Append to end of output expression
Operator ^
Push ^ onto stack
Operator +,-, *, or /
Pop operators from stack, append to output expression until stack empty or top has lower precedence than new operator. Then push new operator onto stack
Open parenthesis
Push ( onto stack, treat it as an operator with the lowest precedence
Close parenthesis
Pop operators from stack, append to output expression until we pop an open parenthesis. Discard both parentheses.

3. Checking for Balanced (), [], {}
The contents of a stack during the scan of an expression that contains the balanced delimiters {[()]}
a{b[c(d+e)/2 – f] +1}

4. Evaluating Postfix Expression
The stack during the evaluation of the postfix expression
a b + c / when a is 2, b is 4 and c is 3

5. The Program Stack
The program stack at 3 points in time; (a) when main begins execution; (b) when methodA begins execution, (c) when methodB begins execution.

6. Circular Linked Implementations of a Queue
A circular linked chain with an external reference to its last node that (a) has more than one node; (b) has one node; (c) is empty.

7. Array-Based Implementation of a Queue
An array that represents a queue without shifting its entries: (c) after several more additions & removals; (d) after two additions that wrap around to the beginning of the array

8. Binary Trees
If a binary tree of height h has all leaves on the same level h and every nonleaf in a full binary tree has exactly two children
A complete binary tree is full to its next-to-last level
Leaves on last level filled from left to right

9. Binary Trees
Total number of nodes n for a full tree can be calculated as:
The height of a binary tree with n nodes that is either complete or full is log2(n + 1)

10. Traversals Exercise
The order of these nodes being visited using 4 different traversal methods

11. Answer In this binary tree, D = node, L = left, R = right
Preorder (DLR) traversal yields: A, H, G, I, F, E, B, C, D
Postorder (LRD) traversal yields: G, F, E, I, H, D, C, B, A
In-order (LDR) traversal yields: G, H, F, I, E, A, B, D, C
Level-order traversal yields: A, H, B, G, I, C, F, E, D

12. Binary Search Trees
Two binary search trees containing the same names as the tree in previous slide

13. Removing an Entry, Node Has Two Children
Node N and its subtrees; (a) entry a is immediately before e, b is immediately after e; (b) after deleting the node that contained a and replacing e with a.

14. Removing an Entry, Node Has Two Children
Node N and its subtrees; (a) entry a is immediately before e, b is immediately after e; (b) after deleting the node that contained a and replacing e with a.

15. Removing an Entry, Node Has Two Children
(a) A binary search tree; (b) after removing Chad;

16. Heaps A complete binary tree Maxheap Minheap
Nodes contain Comparable objects
Each node contains no smaller (or no larger) than objects in its descendants
Maxheap
Object in a node is ≥ its descendant objects. Root node contains the largest data
Minheap
Object in a node is ≤ descendant objects
Root node contains the smallest data

17. Using an Array to Represent a Heap
When a binary tree is complete
Can use level-order traversal to store data in consecutive locations of an array
Enables easy location of the data in a node's parent or children
Parent of a node at i is found at i/2 (unless i is 1)
Children of node at i found at indices 2i and 2i + 1

18. Using an Array to Represent a Heap
Fig 1. (a) A complete binary tree with its nodes numbered in level order; (b) its representation as an array.

19. In Figure 1, the steps in adding 85 to the maxheap
Adding an Entry
In Figure 1, the steps in adding 85 to the maxheap
Begin at next available position for a leaf
Follow path from this leaf toward root until find correct position for new entry

20. Removing the Root To remove a heap's root
Replace the root with heap's last child
This forms a semiheap
Then use the method reheap
Transforms the semiheap to a heap

21. The steps in adding 20, 40, 30, 10, 90, and 70 to a heap.
Creating a Heap
The steps in adding 20, 40, 30, 10, 90, and 70 to a heap.

22. Creating a Heap The steps in creating a heap by using reheap.
More efficient to use reheap than to use add
The steps in creating a heap by using reheap.

23. Breadth-First Traversal
(ctd.) A trace of a breadth-first traversal for a directed graph, beginning at vertex A.

24. Depth-First Traversal
A trace of a depth-first traversal beginning at vertex A of the directed graph

25. Shortest Path in an Weighted Graph
Finding the cheapest path from vertex A to vertex H in the weighted graph

26. (a) A directed graph and (b) its adjacency matrix.
The Adjacency Matrix
(a) A directed graph and (b) its adjacency matrix.

27. The Adjacency Matrix Adjacency matrix uses fixed amount of space
Depends on number of vertices
Does not depend on number of edges
Typically the matrix will be sparse
Presence of an edge between two vertices can be known immediately
All neighbors of a vertex found by scanning entire row for that vertex

28. Adjacency lists for the directed graph
The Adjacency List
Adjacency lists for the directed graph

29. The Adjacency List
Represents only edges that originate from the vertex
Space not reserved for edges that do not exist
Uses less memory than corresponding adjacency matrix
Thus more often used than adjacency matrix