In the story, the rising action is when-

It's a section of the plot leading up to the climax. Rising action is where characters and conflict are developed

  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.