I decided to try splitting the second-story branches for better coverage. The idea is to use ropes from the first level branches on upward, tangled to tension, to emulate the finer branches that would hold the leaves.
Just to get an idea of what the canopy might look like, I fiddled around with making 4"x6" (100mm x 150mm) rectangular "leaves"—about the size I could expect from using the slats from palettes. I just applied them in a hemispherical dome. I started with a modest number that could be easily realized—some 1,900 leaves.
I thought I'd go a little crazy and see how dense the shadow would get if I pushed it to 11,000 leaves. I would guess this would be a very optimistic goal to achieve, but not complete out of reach.