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If you decide to cover your deck, it will cut down on maintenance and the effects of inclement weather conditions. This will increase the lifespan of the deck and open up the possibilities of what you can do with the it during less than sunny conditions.
A covered roof will cost a bit more to install because it requires additional decking materials, adding on to the framing, attaching the covering onto the structure of your home and potentially getting a permit from the city. The decking professional will be able to consult on how the calculated cost will be affected by such factors.
Screening in the deck is the most expensive because of the additional materials and labor involved. Not only will you have to cover the top of the deck, but then you have to cover all three sides with screening material. That means adding a door to access your yard, plus the tools and time involved to add the screen material onto the frame of the deck. It's intensive, but it protects your deck from pests and climate conditions better than just a roof.