I started thinking of leaves that grow oppisite each other like yours and here is what came to mind:

At first I thought it was hackberry, but the leaves are too narrow and pointed on your tree. Here is the link to the hackberry tree photo.
http://images.google.com/imgres?imgurl=http://homepage.mac.com/cohora/plants/image/hackberry.jpg&imgrefurl=http://homepage.mac.com/cohora/plants/hackberry.html&h=480&w=360&sz=28&hl=en&start=11&tbnid=pN9uUz1x9b4BQM:&tbnh=129&tbnw=97&prev=/images%3Fq%3DHackberry%2BTree%26gbv%3D2%26svnum%3D10%26hl%3Den%26sa%3DG

Here is the leaf from American Elm:
http://images.google.com/imgres?imgurl=http://www.sfrc.ufl.edu/4h/AmericanElm/AmericanElmLeafWeb.jpg&imgrefurl=http://www.sfrc.ufl.edu/4h/AmericanElm/AmericanElm.htm&h=386&w=500&sz=51&hl=en&start=9&tbnid=IlMPIFCAyT23zM:&tbnh=100&tbnw=130&prev=/images%3Fq%3DAmerican%2BElm%26gbv%3D2%26svnum%3D10%26hl%3Den%26sa%3DG

But, it isn't an elm either.

Then I came upon a photo of Chinese pistache leaves:

http://www.ci.arlington.tx.us/park/forestry/images/trees/Chinese%20Pistache%20Leaves.jpg

These are more narrow and shaped like your leaves. Do the seed form in brown clusters in the fall? Is the form of the tree more rounded? You probably have a female tree since it has the seeds. I am glad we only have a male tree - pretty fall color, but no seeds.

Golden Rain tree??? I hear they volunteer like crazy.PJ

I don't know what the seeds look like on the tree. I just found out all the hundreds of 'weeds' in my yard were coming from the seeds on the tree from a neighbor. I never paid attention to this tree, I don't know if it had the golden raintree 'blossoms' or seed 'lanterns'.

Marianne

I have a Chinese Pistachio in my front yard and a bunch of Hackberries on the side. I'm 98% positive that's not a Chinese Pistachio, which drops a lot of round seeds in the fall, but they don't sprout much. From your description of 1000's of seedlings it sounds like a Hackberry, but the leaves on mine look smaller and paler, and hang kind of listlessly in clumps. I can always tell Hackberries by their warty bark, and their penchant to grow everywhere. The only thing worse is Ailanthus (Tree of Heaven).

I think its an Elm tree.
Very few of those seedlings actually take hold and live.
You can wait for most of them to die in the coming heat or pull 'em up now.

Not an elm or hackberry b/c it has compound leaves. Looks like you're showing part of the large compound leaf of golden rain tree. Sprouts everywhere and grows in any soil. Beware...

It is a Rain tree (China lantern, Chinese lantern, Lantern tree).
Those of us who have them know them well. In the southern part of Texas they are almost a weed and very fast grower. They are related to the Chinaberry.
They do make a pretty tree with the yellow blooms and pinkish lanterns. Be ready to pull out "lots" of seedlings.