Raised beds are great, and you'll certainly have better soil that way, but - arrgh - the tree roots will still find their way to the water and goodies in those raised beds eventually. I'd start out with maybe 4 beds right in the middle of the open area there, as far from the trees as you can get.
Don't forget to make them narrow enough so you can reach to the middle of the bed without stepping into the bed. We made our raised beds at the school garden 4ft. x 8ft. and the little kids couldn't reach, so we put in two more that were 3ft. x 10ft. and they're much better for everyone!
