There is another tree growing under the bark of my dappled willow tree. How do I get rid of this tree I don't want and is it going to kill my dappled willow tree.

That's what the inside of a tree looks like. Just one tree with some serious problems.

There were a couple of branches that were growing out the side of the tree. The leaves didn't match the dappled willow! So what do I do to save the dappled willow?

Were they a completely different shape? If they looked like the Willow only green, then they ARE the Willow only reversions to "normal". The 'trunk' on the right looks like live tissue growing around dead tissue from a large wound. Can you get a picture of the whole tree and maybe the branch with the different leaves?

This tree is grafted at the top of the trunk so if branches were growing from the side of the trunk, they were most likely the rootstock. Is there anything growing from the top of the trunk? That's the dappled willow. If the trunk isn't too damaged, it will heal with time.

You could start some cuttings but they would grow into a dappled willow bush without a main trunk.

The split is open to about half way around the trunk with a small split on the other side. There is the dappled willow at the top of the tree. It's dark here so I can't take a picture until the morning of the whole tree.