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The minimum cardinality for both sides of the relationship will be 1, because each student will certainly take some courses and - each course will be attended by some students On the other hand, if we keep courses in the database that are no longer actively taught for some reason, then the minimum cardinality on the student side of the takes/is-taken-by relationship will be 0 Figure 8-1 shows a data model for the entities and relationships we have been discussing, using one of many standard approaches for graphically representing the entity-relationship diagram Figure 8-1 was created using Microsoft Visio The rectangles represent entities, and the label in the upper portion of an entity rectangle specifies the identifier, or key, for the entity type For dormitories, for example, the dorm name is the identifier; the name of the dorm distinguishes the record of one dorm from that of another.

public class TestContains { static public void main(String[] args) { BinaryTree<String> e = new BinaryTree<String>("E"); BinaryTree<String> g = new BinaryTree<String>("G"); BinaryTree<String> h = new BinaryTree<String>("H"); BinaryTree<String> i = new BinaryTree<String>("I"); BinaryTree<String> d = new BinaryTree<String>("D", null, g); BinaryTree<String> f = new BinaryTree<String>("F", h, i); BinaryTree<String> b = new BinaryTree<String>("B", d, e); BinaryTree<String> c = new BinaryTree<String>("C", f, null); BinaryTree<String> a = new BinaryTree<String>("A", b, c); System.out.printf("a: %s%n", a); System.out.println("a.contains(\"H\") = " + a.contains("H")); System.out.printf("b: %s%n", b); System.out.println("b.contains(\"H\") = " + b.contains("H")); System.out.printf("c: %s%n", c); System.out.println("c.contains(\"H\") = " + c.contains("H")); } }

The output is:

The labels in the lower portion of the entity rectangles represent the other attributes of the entity The dorm entity includes information for each dorm about the total number of rooms in the dorm, the number of vacant rooms in the dorm, and the room rental rate for the dorm..

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a: [A, B, D, G, a.contains("H") b: [B, D, G, E] b.contains("H") c: [C, F, H, I] c.contains("H") E, C, F, H, I] = true = false = true

The subtrees b and c are shown in Figure 11.22. The tree a contains the element H. The subtree b does not contain the element H. The subtree c does contain the element H.

Figure 8-1 Example entity-relationship (E-R) diagram The lines represent relationships between entities, and the marks at the ends of the lines represent the cardinalities A circle at the end of a line means that the relationship is optional with respect to that entity; a bar at the end of a line means that only one instance of that entity type may participate in an instance of the relationship; and a crows foot means that many instances of that entity type may participate in an instance of the relationship For instance, a single department offers one or many courses; every department offers at least one course Also, a dorm may be associated with one or many students, and a student may be associated with no dorm, or with one dorm.

IMPLEMENTATIONS OF THE TRAVERSAL ALGORITHMS The iterator that is returned by the iterator() method follows the preorder traversal algorithm (Algorithm 11.3 on page 208) to traverse the binary tree. The following modification of the BinaryTree class implements all four of the binary tree traversal algorithms. EXAMPLE 11.23 Implementing the Four Traversal Algorithms

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Some students are commuters who will not be associated with a dorm, but if a student is associated with a dorm, the student is associated with at most one dorm These processes of defining entities, their attributes, and the relationships among entities are effective for most entities and relationships There are a few more special cases, however, that come up often enough to require some additional discussion Some entities belong in the database only if another entity is already part of the database For instance, we would include dependents of a professor in the database only if the professor were already included If a professor leaves the university, the professor s information would be removed from the database, and it would no longer make sense to store information about the professor s dependents, either An entity type such as Dependent is called a weak entity.

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