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1.6 If you were offered a job with Microsoft and permitted to choose between working on operating systems, database products, or applications products like Word or Excel, which would you choose, and why 1.7 Whom do you believe should be credited as the inventor of the modern computer 1.8 What applications of computing seem to you to be unethical What are some principles you can declare with respect to the ethical and unethical use of computers and software 1.9 List some important ways in which computing has contributed to the welfare of humanity. Which people, if any, have suffered from the advance of computing technology

101 Proof of Theorem 101 on page 187: If there were no path from a given node x to the root of the tree, then the definition of tree would be violated, because to be an element of the tree, x must be the root of some subtree If there were more than one path from x back to the root, then x would be an element of more than one distinct subtree That also violates the definition of tree, which requires subtrees to be disjoint Proof of Theorem 102 on page 188: If the tree is empty, then its height is h = 1 and the number of nodes n = 0 In that case, the formula is correct: n = (d (h)+1 1)/(d 1) = (d ( 1)+1 1)/(d 1) = (d 0 1)/(d 1) = (1 1)/(d 1) = 0.

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If the tree is a singleton, then its height is h = 0 and the number of nodes n = 1 In that case, the formula is again correct: n = (d (h)+1 1)/(d 1) = (d (0)+1 1)/(d 1) = (d 1)/(d 1) = 1 Now assume that the formula is correct for any full tree of height h 1, where h 0 Let T be the full tree of height h Then by definition, T consists of a root node and a set of d subtrees And since T is full, each of its d subtrees has height h 1 Therefore, by the inductive hypothesis, the number of nodes in each subtree is nS = (d (h 1)+1 1)/(d 1) = (d h 1)/(d 1).

DEFINITION OF AN ALGORITHM An algorithm is a method for solving a class of problems While computer scientists think a lot about algorithms, the term applies to any method of solving a particular type of problem The repair manual for your car will describe a procedure, which could also be called an algorithm, for replacing the brake pads The turn-by-turn travel instructions from MapQuest could be called an algorithm for getting from one place to another EXAMPLE DESIGNING A STAIRCASE You may be surprised, as we were, to know that every staircase must be custom-designed to fit the circumstances of total elevation (total rise ) and total horizontal extent (total run ) Figure 2-1 shows these dimensions If you search the web, you can find algorithms methods for designing staircases.

Thus, the total number of nodes in T is n = 1 + d nS dh 1 = 1 + d ------------d 1 d 1 dh + 1 d = ----------- + -------------------d 1 d 1.

= ----------------------

To make stairs fit a person s natural gait, the relationship of each step s rise (lift height) to its run (horizontal distance) should be consistent with a formula Some say the following formula should be satisfied: (rise * 2) + run = 25 to 27 inches Others say the following simpler formula works well: rise + run = 17 to 18 inches Many say the ideal rise for each step is 7 in, but some say outdoor steps should be 6 in high because people are more likely to be carrying heavy burdens outside In either case, for any particular situation, the total rise of the staircase will probably not be an even multiple of 6 or 7 in Therefore, the rise of each step must be altered to create a whole number of steps These rules lead to a procedure for designing a staircase.

Thus, by the Principle of Mathematical Induction (see page 321), the formula must be correct for all full trees of any height. 10.3 Proof of Corollary 10.1 on page 188: This proof is purely algebraic: dh + 1 1 n = -------------------d 1 n d 1 = dh + 1 1 dh + 1 = n d 1 + 1 = nd n + 1 h + 1 = log d nd n + 1 h

Proof of Corollary 10.2 on page 188: Let T be a tree of any order d and any height h. Then T can be embedded into the full tree of the h+1 1 same degree and height. That full tree has exactly d -------------------- nodes, so its subtree T has at most that d 1 many nodes.

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