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The next corollary applies Corollary 11.1 together with the fact that the full binary tree of height h has more nodes than any other binary tree of height h. Corollary 11.3 In any binary tree of height h, h + 1 n 2h+1 1 and lg n h where n is the number of its nodes. n 1
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IDENTITY, EQUALITY, AND ISOMORPHISM In a computer, two objects are identically equal if they occupy the same space in memory, so they have the same address. In other words, there really only one object, but with two different names. That meaning of equality is reflected in Java by the equality operator. If x and y are references to objects, then the condition (x == y) will be true only if x and y both refer to the same object. But the normal concept of equality in mathematics is that the two things have the same value. This distinction is handled in Java by the equals() method, defined in the Object class (see 4) and thus inherited by every class. As defined there, it has the same effect as the equals operator: x.equals(y) means x == y. But that equals() method is intended to be overridden in subclasses so that it will return true not only when the two objects are identically equal, but also when they are separate objects that are the same in whatever sense the class designer intends. For example, x.equals(y) could be defined to be true for distinct instances x and y of Point class if they have the same coordinates. EXAMPLE 11.6 Testing Equality of Strings
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public class TestStringEquality { static public void main(String[] args) { String x = new String("ABCDE"); String y = new String("ABCDE"); System.out.println("x = " + x); System.out.println("y = " + y); System.out.println("(x == y) = " + (x == y)); System.out.println("x.equals(y) = " + x.equals(y)); } }
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The output is:
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x = ABCDE y = ABCDE (x == y) = false x.equals(y) = true Here, the two objects x and y (or, more precisely, the two objects that are referenced by the reference variables x and y) are different objects, occupying different memory locations, so they are not identically equal: (x == y) evaluates to false at line 7. But they do both have the same contents, so they are mathematically equal, and x.equals(y) evaluates to true at line 8.
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The distinction between identical equality and mathematical equality exists in Java only for reference variables (i.e., only for objects). For all variables of primitive types, the equality operator tests for mathematical equality.
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Data structures have both content and structure. So it is possible for two data structures to have equal contents (i.e., have the same contents) but be organized differently. For example, two arrays could both contain the three numbers 22, 44, and 88, but in different orders. EXAMPLE 11.7 Testing Equality of Arrays
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public class TestArraysEquality { public static void main(String[] args) { int[] x = { 22, 44, 88 }; int[] y = { 88, 44, 22 }; ch02.ex02.DuplicatingArrays.print(x); ch02.ex02.DuplicatingArrays.print(y); System.out.println("Arrays.equals(x, y) = " + Arrays.equals(x, y)); Arrays.sort(x); Arrays.sort(y); ch02.ex02.DuplicatingArrays.print(x); ch02.ex02.DuplicatingArrays.print(y); System.out.println("Arrays.equals(x, y) = " + Arrays.equals(x, y)); } }
The output it:
{22, 44, 88} {88, 44, 22} Arrays.equals(x, y) = false {22, 44, 88} {22, 44, 88} Arrays.equals(x, y) = true This shows that the java.util.Arrays.equal() method requires not only the same contents for
arrays to be equal, but also in the same order, as would be expected.
Equality is a weaker relation than identity: Identical objects are always equal, but equal objects may not be identical; they could be distinct. Equality of data structures means the same structure and the same contents in the same order. A weaker kind of reflexive relation is isomorphism. Two data structures are isomorphic if they have the same structure. This concept is used when the data part of the data structure is irrelevant. Two arrays are isomorphic if they have the same length. Two trees are isomorphic if one tree can be rearranged to match the other. More formally, T1 is isomorphic to T2 (sometimes written T1 T2 ) if there is a one-to-one mapping (an isomorphism) between them that preserves parent-child relationship between all nodes. EXAMPLE 11.8 Isomorphic Trees
As unordered trees, Tree 1 and Tree 2 in Figure 11.7 are isomorphic, but not equal. However, Tree 3 is not isomorphic to either of the other two trees because it has only three leaves; the other two trees each have four leaves:Tthat s a different structure. That distinction leads fairly easily to a formal deduction that there is no isomorphism between Tree 1 and Tree 3. As ordered trees, Tree 1 is not isomorphic to Tree 2 because their roots left-most subtrees have different sizes. The left-most subtree in Tree 1 has three nodes, while that of Tree 2 has only two nodes. That distinction also leads fairly easily to a formal deduction that no isomorphism between Tree 1 and Tree 2 can exist.
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