I'm new to this so maybe I missed something, but I don't see a location in the question or profile, so many of the recommendations that have been given may be invalid. Most things mentioned would not grow here in zone 4. You need to know your hardiness zone if you're going to buy plants from somewhere other than a local nursery. Even then, southern-based Home Depot sends plants that do not survive Minnesota winters to sell up here. Even they don't get it. If you plant something that dies over the first cold winter, you've lost money, the plant, and most importantly, the year(s) to grow. Really and truly, the best thing to do is visit a local nursery. People that work in nurseries love plants and know your local climate and soil. If you want to order elsewhere, check davesgarden.com first. The Garden Watchdog tab is folks rating their experiences. For example, Michigan bulb company has long been know as the worst for mailing dead and dying plants or calling chunks of dirt bulbs. Also, any tree that is mailed is going to probably be 10 years from being big enough to provide privacy above a fence.
That being said, what I think would be perfect, if it would grow in the area, is a tree lilac. I have two varieties, a Japanese tree lilac in my back yard and an Ivory Silk one in my front yard. One is growing like an over-sized shrub, the other is trimmed up like a tree, you can keep them whichever way you like. I have heavy clay soil and they have grown very well. Southern exposure would be good, the more sun they get, the more beautiful blooms they have in the spring. The roots are well-behaved too - very wise to be concerned about that.
Whatever you plant, be patient. The first year a tree settles in, the second year it grows roots, the third year it grows. I've been disappointed with skimply little trees, then have them shoot up nicely in 3rd and 4th years.