A binary tree is full if all of its vertices have either zero or two children. Let Bn denote the number of full binary trees with n vertices.

(a) By drawing out all full binary trees with 3, 5, or 7 vertices, determine the exact values of B3, B5, and B7. Why have we left out even numbers of vertices, like B4?
(b) For general n, derive a recurrence relation for Bn.
(c) Show by induction that Bn is (2n).

Write a program that uses a random number generator to generate a two digit positive integer and allows the user to perform one or more of the following operations: a. double the number. b. reverse the digits of the number. c. raise the number to the power of 2, 3, or 4. d. sum the digits of the number. e. if the number is a two-digit number, then raise the first digit to the power of the second digit. f. if the number is a three-digit number and the last digit is less than or equal to 4, then raise the first two digits to the power of the last digit. after performing an operation if the number is less than 10, add 10 to the number. also, after each operation determine if the number is prime. each successive operation should be performed on the number generated by the last operation. your program should not contain any global variables and each of these operations must be implemented by a separate function. also, your program should be menu driven. 7. (fraction calculator) write a program that

Read “suburban homes construction project” at the end of chapters 8 and 9 (in the textbook) and then develop a wbs (work breakdown structure) in microsoft excel or in microsoft word (using tables)

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