Berkovich spaces and tropical intersections

Berkovich spaces are a foundational object in non-Archimedean geometry, and can be thought of as infinite glueings of tropicalizations.  
After introducing the Berkovich line (an infinitely branching tree), minimal skeleta, and faithful tropicalizations, I will show how Berkovich theory can be used to characterize possible images of intersection points under tropicalization.  
There will be many examples, and no previous knowledge of Berkovich spaces is required!