I think tree branches will be a rather large problem for these drones since you need a pretty high resolution 3D sensor to accurately detect them.

GPS is most certainly not enough for navigation next to a building since you can hardly get an accuracy below 5m. I would bet on navigation by computer vision and 3D sensors.

Another interesting aspect of this is the choice of location where to drop the package. You certainly would not appreciate finding your emergency math textbook in the middle of your flower bed. This could for example be solved by setting up delivery spots in advance, or perhaps by using human judgement through something like Amazon's Mechanical Turk.