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These are mutually recursive implementations of the hyperbolic sine and cosine functions:
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public static double s(double x) { if (-0.005 < x && x < 0.005) { return x + x*x*x/6; // basis } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 + x*x/2; // basis } return 1 + 2*s(x/2)*s(x/2); // recursion }
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This is a recursive implementation of the tangent function:
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public static double t(double x) { if (Math.abs(x) < 0.5e-10) { return x + x*x/3 + x*x*x*x/5; // basis } double tx2 = t(x/2); return 2*tx2/(1 - tx2*tx2); // recursion }
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This is a recursive evaluation of a polynomial function:
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public static double p(double[] a, double x) { // returns a[0] + a[1]*x + a[2]*x*x + ... return p(a, x, 0); } private static double p(double[] a, double x, int k) { // returns a[k] + a[k+1]*x + a[k+2]*x*x + ... if (k == a.length) { return 0.0; // basis } return a[k] + x*p(a, x, k+1); // recursion }
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A tree is a nonlinear data structure that models a hierarchical organization. The characteristic features are that each element may have several successors (called its children ) and every element except one (called the root ) has a unique predecessor (called its parent ). Trees are common in computer science: Computer file systems are trees, the inheritance structure for Java classes is a tree, the run-time system of method invocations during the execution of a Java program is a tree, the classification of Java types is a tree, and the actual syntactical definition of the Java programming language itself forms a tree. TREE DEFINITIONS Here is the recursive definition of an (unordered) tree: A tree is a pair (r, S), where r is a node and S is a set of disjoint trees, none of which contains r. The node r is called the root of the tree T, and the elements of the set S are called its subtrees. The set S, of course, may be empty. The restriction that none of the subtrees contains the root applies recursively: r cannot be in any subtree or in any subtree of any subtree. Note that this definition specifies that the second component of a tree be a set of subtrees. So the order of the subtrees is irrelevant. Also note that a set may be empty, so (r, ) qualifies as a tree. This is called a singleton tree. But the empty set itself does not qualify as an unordered tree. EXAMPLE 10.1 Equal Unordered Trees
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The two trees shown in Figure 10.1 are equal. The tree on the left has root a and two subtrees B and C, where B = (b, ), C = (c, {D}), and D is the subtree D = (d, ). The tree on the right has the same root a and the same set of subtrees {B, C} = {C, B}, so (a, {B, C}) = (a, {C, B}).
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Figure 10.1 Equal trees
The elements of a tree are called its nodes. Technically, each node is an element of only one subtree, namely the tree of which it is the root. But indirectly, trees consist of nested subtrees, and each node is considered to be an element of every tree in which it is nested. So a, b, c, and d are all considered to be nodes of the tree A shown Figure 10.2. Similarly, c and d are both nodes of the tree C.
CHAP. 10]
TREES
The size of a tree is the number of nodes it contains. So the tree A shown in Figure 10.2 has size 4, and C has size 2. A tree of size 1 is called a singleton. The trees B and D shown here are singletons. If T = (x, S) is a tree, then x is the root of T and S is its set of subtrees S = {T1, T2, . . ., Tn}. Each subtree Tj is itself a tree with its own root rj . In this case, we call the node r the parent of each node rj , and we call the rj the children of r. In general, we say that two nodes are adjacent if one is the parent of the other. Figure 10.2 Subtrees A node with no children is called a leaf. A node with at least one child is called an internal node. A path in a tree is a sequence of nodes (x0, x1, x2, . . ., xm) wherein the nodes of each pair with adjacent subscripts (xi 1, xi) are adjacent nodes. For example, (a, b, c, d) is a path in the tree shown above, but (a, d, b, c) is not. The length of a path is the number m of its adjacent pairs. It follows from the definition that trees are acyclic, that is, no path can contain the same node more than once. A root path for a node x0 in a tree is a path (x0, x1, x2, . . ., xm) where xm is the root of the tree. A root path for a leaf node is called a leaf-to-root path. Theorem 10.1 Every node in a tree has a unique root path. For a proof, see Problem 10.1 on page 194. The depth of a node in a tree is the length of its root path. Of course, the depth of the root in any tree is 0. We also refer to the depth of a subtree in a tree, meaning the depth of its root. A level in a tree is the set of all nodes at a given depth. The height of a tree is the greatest depth among all of its nodes. By definition, the height of a singleton is 0, and the height of the empty tree is 1. For example, the tree A, shown in Figure 10.2, has height 2. Its subtree C has height 1, and its two subtrees B and D each have height 0. A node y is said to be an ancestor of another node x if it is on x s root path. Note that the root of a tree is an ancestor of every other node in the tree. A node x is said to be a descendant of another node y if y is an ancestor of x. For each node y in a tree, the set consisting of y and all its descendants form the subtree rooted at y. If S is a subtree of T, then we say that T is a supertree of S. The path length of a tree is the sum of the lengths of all paths from its root. This is the same as the weighted sum, adding each level times the number of nodes on that level. The path length of the tree shown here is 1 3 + 2 4 + 3 8 = 35. EXAMPLE 10.2 Properties of a Tree
The root of the tree shown in Figure 10.3 is node a. The six nodes a, b, c, e, f, and h are all internal nodes. The other nine nodes are leaves. The path (l, h, c, a) is a leaf-to-root path. Its length is 3. Node b has depth 1, and node m has depth 3. Level 2 consists of nodes e, f, g, and h. The height of the tree is 3.
Nodes a, c, and h are all ancestors of node l. Node k is a descendant of node c but not of node b. The subtree rooted at b consists of nodes b, e, i, and j.
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