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?
The rules do not explicitly address this situation, but I would rule that Damaging the tree will not damage the staff, however, destroying it will destroy the staff as well.
I will first discuss why and then follow up with a discussion of possible pitfalls.
How to damage or destroy magical items
Based on the DMG, damaging most magical items is fairly easy; they are objects and are resistant to all damage (DMG p. 141):
p141 Most magic items are objects of extraordinary artisanship, assembled from the finest materials with meticulous attention to detail. Thanks to this combination of careful crafting and magical reinforcement, a magic item is at least as durable as a regular item of its kind. Most magic items, other than potions and scrolls, have resistance to all damage. Artifacts are practically indestructible, requiring extreme measures to destroy.
As per rules for objects (DMG p. 246), a staff made from wood would likely be a small object made of wood (15 AC and 3d6 HP), with resistance to all damage since it is a magical item.
What happens to the staff's statistics when it transforms
Tree Form. You can use an action to plant one end of the staff in fertile earth and expend 1 charge to transform the staff into a healthy tree. The tree is 60 feet tall and has a 5-foot-diameter trunk, and its branches at the top spread out in a 20-foot radius. The tree appears ordinary but radiates a faint aura of transmutation magic if targeted by detect magic.
Does the staff retains its old statistics or do its statistics change into those of the new form.
This is not explicitly stated by the feature, but the rules for objects suggest that they are based on the physical properties of the object. Therefore the tree should have its own statistics. Let's say it's a large wooden object (15 AC, 5d10 HP).
Is the tree also resistant to all damage?
Since it is no longer
object[s] of extraordinary artisanship, assembled from the finest materials with meticulous attention to detail
tree with a faint magical aura
It does not retain resistance to all damage.
Are the HP pools of the tree and the staff separate?
Again the rules do not provide explicit guidance in this area. There are similar effects that affect creatures, both polymorphed creatures and Wildshaped druids have HP pools separated between their own form and transformed form.
I already sided with the reasoning that the statistics of the object are based on its physical properties, so to remain consistent, I would rule that they have separate HP pools.
If we take guidance from wild shape or polymorph, the tree would always be fully healthy. The staff's damage would be retained between transformations and would have to be repaired by other means.
If the tree is destroyed, can it still be transformed back into the staff?
Here rules are relatively clear, the tree is an object that will be destroyed if it's HP is reduced to 0, it will be destroyed, and the wielder of the staff will no longer be able to transform it back.
Therefore I would rule that:
Damaging the tree will not damage the staff, however, destroying it will destroy the staff as well.
Additional concerns and possible alternative rulings
I will not go in too much detail as I believe that the ruling mentioned above is the one that is most supported by the rules, but as the rules are not explicit here, other explanations are possible:
If the staff's characteristics stay the same after the transformation, then the damage done to the tree should be retained over transformations and vice versa.
If the tree is considered a creature rather then an object: the rules are not super clear on this, see Are plants creatures or objects? then of course rules for creatures would apply. Also, a question that touches on this is How can player characters, creatures and NPCs heal plants?
While supported in the rules over my years of playing D&D 5e, I have not seen magical items taking damage as objects seen used in play ever. In all the games I was part of, they have always been considered virtually indestructible except "plot reasons" Introducing this rule might completely transform the game you are playing.
Generally, players consider their magic items very precious and taking them away might lead to problems at the table
Although the rules don't say anything about this specific circumstance, the DM could apply the same rules as the Druid's Wild Shape.
When you transform, you assume the beast’s hit points and Hit Dice. When you revert to your normal form, you return to the number of hit points you had before you transformed. However, if you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form. For example, if you take 10 damage in animal form and have only 1 hit point left, you revert and take 9 damage. As long as the excess damage doesn’t reduce your normal form to 0 hit points, you aren’t knocked unconscious.
The only change could be that it breaks if it drops to 0 HP.
By RAW no. But thematically, why not?
Staff of the Woodlands reads:
transform the staff into a healthy tree
you return the staff to its normal form
It says nothing about damage being retained, or even if destroying the tree would destroy the staff.
However as a DM I would focus on the fact that it is a transmutation spell that transforms an object. Damaging the tree can certainly be seen to damage the staff. It is hard to imagine that the tree could be cut down, processed, turned into matchsticks, used to create a fire, then reverted back to a perfectly fine staff.