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). 2.14. You are given an array of n elements, and

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Write a loop that reads positive integers from standard input and that terminates when it reads an integer that is not positive. after the loop terminates, it prints out, on a line by itself and separated by spaces, the sum of all the even integers read, the sum of all the odd integers read, a count of the number of even integers read, and a count of the number of odd integers read, all separated by at least one space. declare any variables that are needed. assume the availability of a variable, stdin, that references a scanner object associated with standard input. that is, stdin = new scanner(system.in); is given.

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