Data-Structures-and-Algorithms-in-Java-6th-Edition | Code extracts and notes from the book by Michael | Learning library

The difference in numbers that you see in the recursion trace comes from n/2 in the code. It is an integer division by two. So 13/2 is 6 and 6/2 is 3 and 3/2 is 1, and finally 1/2 is 0. This is what you see when you read the diagram from top to bottom. The diagram shows these entries shifted more and more to the right to represent the depth of the recursion. The function calls itself, and then it calls itself again, ...etc. All of these calls are pending... Each time these calls pass a value for n that is smaller (halved).

Somewhere this stops. It needs to, of course. This stops when the value of n receives the value 0. You see this state at the bottom of the diagram. At that moment, the recursion does not go deeper and starts to unwind, and you need to read the diagram now from bottom back to the top. For n equal to 0, the return value is 1 (return 1).

This value is read by the function execution where n is 1 and where the call power(2, 0) was made and waiting for a return value. It receives 1 as result. This value is then squared (partial * partial) which in this case still is 1. And because n is odd (it is 1 here), there is an additional multiplication *= x. So we actually calculate partial * partial * x, which is 1 * 1 * 2. You'll see this in the diagram.

And this value (2) is returned to the function execution where n is 3 and where the call power(2, 1) was made. It receives 2 as result. This value is then squared (partial * partial) which in this case is 4. And because n is odd (it is 3 here), there is an additional multiplication *= x. So we actually calculate partial * partial * x, which is 2 * 2 * 2. You'll see this in the diagram.

I hope you see the pattern. The function executions that are pending for a result, all get their value back from the recursive function call they made, do some multiplication with it and return that, in turn, to their caller.

And so the recursion backtracks to the top, providing the end result to the top-level call of power(2, 13).