I have both hawks and eagles. My chicken yard is not fenced above, but the coop is raised off the ground which gives the birds a hidey-place when a predator flies over. My rooster is really good at warning the hens and they all huddle up together out of harm's way. If your coop is flush to the ground, I would suggest putting up some sort of protective roof, which will also give them a spot to be outside when it is raining.
I have one young fruit tree inside the yard, which I currently have wrapped with a chicken wire cage. I think once the trunk has bulked up and the branches are higher than a hen can jump, I will remove the cage. In a secondary yard, I have a mature hemlock which they just scratch around and don't bother the bark (so far).
My chicken yard is a repurposed corral, 4 board fence, so perhaps a bit taller than 4'. We attached woven wire to the bottom 3 boards, buried partway into the ground. Four of my 10 hens regularly fly/flop in and out of the chicken yard, but I don't mind them being loose. The rooster used to also get out, but he is now too heavy (good thing, he's meaner than dirt).
Size: I am running 11 birds in about 800 square feet of yard. I started with baby chicks last March and by the end of summer, they had eaten the once lush pasture grass down to bare dirt. We recently added a secondary yard that we open up for a couple hours a day to keep them on fresh grass. And, I let them all out for about an hour per day of free ranging, late afternoon. So far (and I say this cautiously) they have not done any serious damage to my garden beds, although they do make a bit of a mess tossing things around.
Also so far, no predator problems. We have an automatic chicken door on the coop, which closes the door at dusk and opens it at dawn, so the birds are contained during the night when most predators are active. Much as I dislike the rooster, I think he is a really good watchdog of the hens. The neighbors on either side of me have hens but no roosters and they have both lost chickens on a fairly regular basis. My known predators are possum, raccoon, coyote, bobcat, weasel, hawk, eagle, and owl.
Have fun with your chickens, I find them rather amusing. And love the eggs!
