Hi there Janet,
Thank you kindly for all this help. Salix amygdaloides looks like a possibility.
I definitely did not pull them out but cutting them back is a possibility. I am going to have a go at moving the dirt this weekend. When the weather turned cold, I do remember they shriveled up, quickly. I do not have any tree near that matches what was growing. Then again, I bought the house only recently so do not have the full history of what grew here before. The soil where these popped up is extremely poor. The soil I placed over them is of better quality and was going to mix it in. As mentioned though, I am going to move it away so I can have a look at the plants again.
The ground is very dry and the only water that hit this area in the summer was from rain runoff.
Thank you so much for all the help!