Write a recursive method body for the static method...

Write a recursive method body for the static method emptyTreeCount. Note that t is a restores-mode parameter. You can only use the Tree kernel and Sequence kernel methods. For full credit, you must use only one return statement. /** * Reports the number of empty (sub)trees in t. * @ensures emptyTreeCount = [the number of empty (sub)trees in t] */ public static int emptyTreeCount(Tree<T> t) { }

Transcript

00:01 We were asked to answer some questions about sequences generated by newton's method.

00:08 So, the newton's method applied to a differential function, f has a starting value x not and constructs a sequence of numbers, according to their recursion formula, xn plus 1 equals xn minus f of xn over f prime of xn.

00:31 And in part a, we're given a function, f, and we're asked to write their recursion formula.

00:51 So f of x equals x squared minus a, where a is greater than zero.

01:00 And so we have that their recursion formula can be calculated first.

01:06 We need to calculate the derivative.

01:08 So the derivative of f is 2x.

01:13 And so we have their accursion formula for f.

01:16 It's going to be xn plus 1 equals xn minus, x n squared minus a over to x n and this can be rewritten as the factor out or if we divide we get xn minus this becomes xn over 2 plus a over 2 xn and this is equal to after adding like terms we get 1 half x n plus a over 2 x n and this is equal to after adding like terms we get 1 half x n plus a over 2 xn, which can be written as xn plus a over xn divided by 2.

02:52 And in part b, we're given values for the first term of the sequence and a.

02:59 And we're asked to calculate successive terms until the display repeats.

03:08 So we have that the first term is going to be 1, and that a is going to be 3.

03:19 And so by the previous part, we have the recurrence relation is xn plus 1 equals xn plus 3 over xn divided by 2.

03:41 And so we have that x0 is as we're given 1...

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