http://www.smugmug.com/gallery/12641899_Jtqdk

There's a hottub pump housing at the side of my house with a rotted roof. It
is prone to rotting because a fir tree above drops pine needles on the roof,
and these retain moisture which rots the roof. Furthermore, this roof was
made what looks like fence boards, with no roofing tiles on top.

I would like to replace it with a proper roof. I believe this means:

a plywood (what thickness, what quality?)
metal drip edge
tar paper (how about plastic sheet?)
composite roofing tiles

Is there a simpler solution that works just as well?