# Point rank in 2D plane time complexity?

I'm reading about the algorithm of finding the ranks of all points in a 2D plane, I don't understand the time complexity formula for it. It has four steps:

1. Compute the median of x-coordinates of all point, and divide the plane into two half Left, and Right.
2. Recursively do 1. then when there is only 1 point, rank(that point)=0.
3. Sort points by y-coordinate in Left, and Right separately.
4. Update Right.

I understand the idea of these steps, and 3. has complexity $$O(n\log n)$$, but the time complexity formula in my book is

$$T(n)=2T(n/2)+\Theta(n),$$

why the last term is not $$\Theta(n\lg n)$$? That is my current idea is that the $$T(n)=\Theta(n\lg^2n)$$, by applying master's theorem.

I think I may solve it but I'm not sure: In short the sorting can be done in $$\Theta(n)$$, so the book statement, hence my assumption, is not correct. The "Sort points by y-coordinate in Left, and Right" actually only need to merge it, and this step is $$\Theta(n)$$.