Answer. At first blush this appears to be an issue involving "hazard trees" under ORS 90.100(21). A "hazard tree" is one that:
Is located on a rented space in a manufactured dwelling park;
Measures at least eight inches DBH; and
Is considered, by an arborist licensed as a landscape construction professional pursuant to ORS 671.560(Issuance of license) and certified by the International Society of Arboriculture, to pose an unreasonable risk of causing serious physical harm or damage to individuals or property in the near future.
Regarding hazard trees, ORS 90.727(3) (Maintenance of trees in rented spaces) provides that a landlord:
(a) Shall maintain a tree that is a hazard tree that was not planted by the current tenant, on a rented space in a manufactured dwelling park if the landlord knows or should know that the tree is a hazard tree.
(b) May maintain a tree on the rented space to prevent the tree from becoming a hazard tree, after providing the tenant with reasonable written notice and a reasonable opportunity to maintain the tree;
(c) Has discretion to decide whether the appropriate maintenance is removal or trimming of the hazard tree; and
(d) Is not responsible for maintaining a tree that is not a hazard tree or for maintaining any tree for aesthetic purposes.
When the hazard tree legislation was enacted, as best as I can remember, all of us involved in the drafting were looking up, not down. That is to say, we were focusing on large limbs falling on homes, or trees becoming top-heavy from a lack pruning and trimming, and falling over entirely. I do not recall any discussion about the root systems of trees causing dangerous conditions.
However, clearly, if the root system of a tree meeting the definition of a "hazard tree" causes walkways to buckle, it can create a danger to the residents and others using them. Accordingly, under a pretty clear reading of the above-cited statutes, this appears to be a landlord issue that needs to be addressed. As between a resident who does not own the space or the common area walkways, and the landlord who does - and who presumably has the means and insurance to protect against the risk - the financial responsibility rests on the latter. It is a cost of doing business.
However, what if the tree does not meet the definition of a "hazard tree," is located on the resident's space, but has a root system that is encroaching up into common area walkways causing damage? That is a more difficult issue. I say this because if it isn'ta hazard tree, say by girth, then the resident has the trimming and pruning responsibility. But as to that portion of the root system encroaching into the common area, I believe that is the landlord's responsibility. Certainly, the problem did not result from a lack of resident maintenance.
My conclusion is this: The landlord should want the common area walkways safe. If this means cutting out the offending root system, the landlord should do so. Why, because if a resident or their guest falls and injures themselves on a broken walkway, you can be sure the landlord will be sued. Again, it's a cost of doing business, and presumably the reason why smart landlords have good liability insurance.
 "Diameter at breast height" i.e. 4.5 feet. See: http://www.phytosphere.com/treeord/measuringdbh.htm