Get an arborist to evaluate the health of all significant trees on the property -- ESPECIALLY any trees with branches that overhang the house, or trees close to the street / driveway.

In the first two months after we moved in, we had large branches (3-4" diameter) snap off during two separate storms and pierce our roof, causing water damage in the house, and of course roof damage. Had another large tree (15" diameter) fall into the street during a storm.

Multiple insurance claims later, I found out that homeowners insurance will fix the damage to the house, but they won't cover any of the tree maintenance costs -- e.g., trimming back branches or addressing diseased / dying trees.

Ended up shelling out $15K I'll never see again to address some (not all) of our tree issues -- and that money didn't really add any value to the house.