Jeffp51, I make a lot of bows out of red elm (slippery elm) saplings, because I can cut as much as I want in my neighborhood, and all around. It's ok, but as Marc mentioned VERY variable. I've had wonderful wood, and then really very marginal wood from it, even in the same thicket. Some was dense and springy, with thick rings and hearty sapwood, and some was fine ringed, almost punky, and very light. I like making lumpy flatbows out of it, and I believe any eklm is bow wood, but I guess not any tree is....
One thing to remember is that most elms seed early in the year and the seeds blow and volunteer everywhere. So elm spreads fast, and you often find several species together along a waterway, for instance, because of seeds that came out of yards etc. I know I have a patch of volunteers growing in a thicket in a vacant lot near my house that has at least two species growing mixed. On has bigger, rough leathery leaves and leafs out later, makes fewer seeds, and better bows.
That said, the elms with the small heartwood, lots of sapwood, and thick rings do make better bows. I haven't used all species, but rock elm is the best I have used. Any specie found in someone's yard might be found growing wild nearby.