Unless you're making an effort to contain the view within the patio area itself--as you would in an interior room--I'd disregard the advice about having a "focal point" and consider the view to the outside world, the "objet d' vu." Using greenery to frame the view could be a nice feature. The plants could be more or less out of the way and rather than BE a feature, just be the edge to the "windows" that are already there. A climbing vine such as heavenly blue morning glory, moonvine, or Mandevilla or something capable of climbing a cord trellis would work, The "trellis" could be hung on a couple of stout hooks in the wood ceiling members. The hooks could remain permanently but the cord trellis could be lifted off and removed as necessary. As a practical concern, a planter needs to be large enough to support plant life in a healthy manner, but other than that, not having it any larger than necessary would help preserve balcony space. A half-round planter such as the one pictured below could work for the center post. Quarter-round planters are hard to find, so you could use anything that's compatible, and probably narrower, for the outer walls. Some shade tolerant, blooming annuals planted around the outer edge of the planter (begonias, lobelia or something along those lines) would spill over its edge and be a nice touch, too. I like that the rail and pickets have a light/view filtering effect rather than act as a complete screen to the lower portion of the view. If they were mine, I'd prefer not to cover them to any appreciable degree with foliage.

We were in a similar situation. We have a gray stamped concrete patio (block pattern). We put in a light tan color Koolcrete (sp?) that's what it was called when we had it installed. On the link below they call it Frontier Desert Deck (don't know if it's the same product but a different name) It looks similar we have the Sand Buff Color. Looks good, stays cool, cleans up well. It's been installed for approx. 3 1/2 years. I would use it again. Here is a link that might be useful: frontier desert colors (scroll down)

It might be more boring because it doesn't seem to have nearby views/landscaping. There's no planting around the house or fence to speak of. Given that it seems remote for you, maybe it's just spillover space for the occasional large gathering. Or a place to "get away" (fire pit). I'm confident that an arrangement of useful furnishing would make it seem like less of a "blob."

First we need pictures. Second , I would get the tree pruned off your property.