The phrase "largest branch" refers to the biggest division or part of something, usually a company or organization, that has multiple smaller divisions. Full definition
There could easily be a tree or large branch in the road around the next bend so it's important to drive slowly with caution as hitting debris at speed could be fatal. (reddrivingschool.com)
This resulted in large branch offices with high pay outs to the sales managers which ideally should have been agent - servicing locations. (indianexpress.com)
Fallen trees or large branches also are best removed by a professional tree service. (nationwide.com)