I was looking again at the issue of writing breadth-first traversals for rose trees, and in particular the problem explored in Gibbons (2015). The breadth-first traversal here is a traversal in the lensy sense.
When f is specialized to , we get the original rose tree. But what we actually require is much less specific: levels, for instance, only needs Foldable.
The overhead from this approach scraps any benefit, though.