Usage patterns
枝が伸びるfor a branch to grow/extend
The branches of the tree in the yard keep growing longer.
枝が広がるfor branches to spread out
That big tree's branches spread out in every direction.
枝を折るto break off a branch
The child broke a branch off the tree while playing.
〈物〉が枝にとまるfor something (e.g. a bird) to perch on a branch
A small bird is perched on the tree branch.