I second the Christmas lights idea. Make sure you get the big bulbs -- the 7 or 10-watt ones. Then figure out your heating requirement based on the square footage of your cover and the difference in temperatures you need to maintain, and get that many bulbs. A tip would be to roll up the excess material and hold it closed with spring clamps to reduce the exposed area of the cover to slow heat loss.
The problem with trees is that they tend to be tall, so while you can cover them temporarily, a (seasonally) permanent cover is going to have to be very sturdy to keep it from getting blown away during a storm. There's not really any good solution I know of.