Tropical Geometry

In the paper below it describes Proposition 3.2 as “ Let p ∈ C ∩C′ which is a vertex of neither C nor C′. Then the number of intersection points of Ct and C′t whose image under Logt converges to p is exactly equal to m( p).”.

I don’t understand how it is possible for p to be in that set and yet also not be in C nor C’. To me this seems perfectly counterintuitive. I’d appreciate any help on the matter.

As a second question, I’d be interested in the reasoning why the solution to this when d = 3 and r = 1, for example, is 12:

how many irreducible complex algebraic curves of degree d with r double points pass through a generic configuration of d(d+3)/2 − r points?

http://erwan.brugalle.perso.math.cnrs.fr/articles/EMS/TropEMS.pdf