We have a couple different hawks that hang out on our back fence near our oleander occasionally -- laying in wait for an unsuspecting dove, finch or sparrow to alight on one of the oleander branches or our deck railing.
Most recently -- just a couple weeks ago, in fact -- we had a Coopers hawk in our backyard for an hour or so. I do not think he was successful in catching a meal, though. We have also had a red shouldered hawk but that was roughly a year ago; he came by fairly often for about a month and then we stopped seeing him. I guess he moved on.
We, too, have an easement (read: flood plain) behind our home, and although it is not heavily wooded (just scrub) it is nice to have back there. Growing up, almost all of the homes I ever lived in with my parents had a greenbelt or other open space behind the house. My folks have an awesome, heavily wooded greenbelt behind them and the variety and sheer number of birds they get in their backyard (which, to be fair, is much larger than ours) is truly amazing.
If you are not in the Cornell Laboratory of Ornithology's
PROJECT FEEDERWATCH, I highly recommend it. It's an annual, winter-long count of birds in your area and the data is collected and studied by Cornell and other bird study groups to understand bird species distribution, general health of the population and to note changes due to environmental or other factors. It's really a treat and is celebrating its 20th year this year.
My mom's participating in FeederWatch for the first time this year, and is quite excited to report her counts each week. Justin and I have been members off and on for over 10 years -- some years we paid the membership but didn't submit counts because we lived in an apartment or whatnot, but we're participating again this year. It's great for kids, too, once they're old enough to identify common birds; my 2-year-old niece in Dallas knows bluejay, cardinal, mockingbird, dove and ducks -- they live near a pond, so she gets to see the latter fairly often.