If you turn the staff of the woodlands into a tree with its ability, and then cut off a branch of the tree, would it damage the staff?
[RPG] Does taking a branch from the tree produced by the Staff of the Woodlands harm the staff
dnd-5emagic-items
Related Solutions
The item provides:
Objects in the area that aren’t being worn or carried take this damage and are pushed automatically.
Your small cave's walls get pushed away to accommodate the fortress.
As for generalizing, you can't: at least not from an item that has a specific rule. Generally, generalizing (heh!) would need a general premise to start from and I don't believe that there is one - these types of items are either specific about what happens or leave it unsaid and therefore up to the DM.
The Staff of the Woodlands is a druidic focus
As you've pointed out , the PHB gives a few examples of what a druidic focus should be and "a staff drawn whole out of a living tree" is certainty the most applicable option in the list.
Druidic Focus. A druidic focus might be a sprig of mistletoe or holly, a wand or scepter made of yew or another special wood, a staff drawn whole out of a living tree, or a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.
However, that list is not an exhaustive list. "A druidic focus might be" one of those things, but it could also be a something else.
Moreover, one of the options for a druidic focus in the Adventuring Gear table (PHB p150) is simply noted as a "Wooden staff". D&D Beyond is organized slightly differently, but that "Wooden staff" is still listed as a druidic focus.
Staff of the Woodlands does not describe how it is made of so one could question whether it has been "drawn whole out of a living tree". However, no one could reasonably claim that this isn't a wooden staff, given its theme and depiction:
Finally, there is precedent for a magic item that functions as a focus, even though that magic item does not explicitly say it is a focus.
Best Answer
The rules don't say anything about this circumstance. The GM should adjudicate. Personally, I don't think it should be damaged, but any branches taken from the tree would disappear when the staff turns back.