what is the best way to continue these thoughts into one sentence soon after the first real trains appeared

The best way to combine these thoughts into one sentence is: Soon after the first real trains appeared, some engineers began building miniature ones.

Explanation:

I just revised it myself.

  this riddles' answer is library.

  the "leaves" are a reference to the books' pages that the library contains; the branches are a network of libraries that connect multiple libraries' collections, resources, and items in a system of a common database, integrating them; the branches' organization are a useful management tool for coordinating the multiple collections many libraries have.