Answer. The land belongs to the landlord, and with the exception of (a) what a tenant plants on their own space[1], and (b) hazard tree responsibilities under ORS 90.727, it is my opinion that what grows out of the ground is the landlord's responsibility. Thus, the trees are the landlord's responsibility, and certainly, they cannot be left to grow to the point of prohibiting removal of a home - or to put it another way, if the trees are permitted to grow in such a manner as to block homes, it is the landlord's responsibility to deal with the cost of removal, so the home can be relocated.
As for the landlord's liability for costs to move the home, etc., I'm going out on a limb - pardon the pun - but that is really a matter of causation, i.e. why did the landslide occur? You did not state in your question that the tenant was somehow responsible for the slide, so I'm going to assume it was an "Act of God". Most good liability insurance policies do cover "Acts of God" events, and hopefully the landlord has such a policy.
So if the ground moves due to earthquake, flooding, shifting, etc., the cost to relocate the home is probably something the tenant would claim against the landlord. If the insurance carrier provides coverage, then that will be the source of payment to the tenant. If there is no such coverage, the question becomes whether a landlord can be held liable for an event the landlord did not cause, i.e. an "Act of God". Without doing a little research, I don't have a good answer, but that is a good reason for the landlord to want to have a good liability policy.
As I've written before, the landlord is in the business of renting out the ground, and therefor has the incentive to insure against risks to the ground, even those caused by "Acts of God". Tenants do not have an insurable interest in the ground, and accordingly, one would not expect them to assume the risk of something happening to property owned by the landlord.
[1] Your question did not indicate that the trees were planted by the tenant or on the tenant's space. In any event, it seems to me that other than maintaining the trees on one's space (subject to the hazard tree statute) they are still the landlord's responsibility, since they will remain when or if the tenant moves their home out.
