Possibly mulberry? Leaves are quite variable by time and age... even on the same tree. Growth you described is typical of the ones we get from the birds every year or so.

This looks just like the one that came up last year in one of my flower beds. To me, it's a junk tree and it will be poisoned this weekend. I can't ID it though. It is not a tree I grow in my yard. I have mesquites, pecans, ash, and a fruitless mulberry.

I thought mulberry too, which is fine if you want to raise silk worms ... ;-) As a new gardener it was recommended to me as a fast growing tree. But I noticed they always seemed to be covered with caterpillars and so were very raggity looking. I guess a lot of moth caterpillars like them,

Thanks for the responses, folks. I found these two photos of "young mulberry sapling" and "mulberry leaves" and my leaf sure is similar. If it's a mulberry......... it's outta here. It's situated VERY close to a medium size oak surrounded by Aspidistra, so unless it were something really special and ornamental, I'll probably be taking it out.

White Mulberry. Repeatedly asked about.