General:
* Limbs
* Arms
* Twigs (for smaller branches)
* Boughs (for larger branches)
* Sprouts (for new branches)
For a company or organization:
* Divisions
* Departments
* Units
* Offices
For a river:
* Tributaries
* Streams
* Affluents
For a family tree:
* Lines
* Descendants
* Offshoots
For a tree's structure:
* Forking
* Diverging
* Extending
To choose the best synonym, consider the specific meaning of "branches" in your sentence.