I have a 20' x 20' run that starts at 8'+ and ends at 4' high. I have 1X6 "rafters" across the high and low ends. I ran 1/8" cable around the remaining perimeter and across the middle from both directions (I set a post in the center to support the cables). I did all of this to support deer netting across the top of the run. J clips attached the netting to the cables (my chicken wire walls too) and seamed the netting together. keeps the chickens in and the hawks out. The netting is plastic, and has lasted a year so far, but at $14 with plenty of left over I can replace it once a year and not care. I should add I do have electric running at the bottom, 6" up, 4' up, and at the top of the run, so I'm not too concerned with predators climbing up and tearing through the netting, but if you plan on locking up your birds at night anyway you won't need too worry about much anyway. My coop is at the high end, and this lets me work in most of the run without hunching over much (I'm 6'3"). While this does not tell you how high a fence needs to be to keep chickens in, it gives you a cheap way of solving the problem to start with. I think the "roof" cost about $40 (maybe less).
Swamp