If a dog runs around a squirrel running around a tree, is the dog really running around the squirrel?

If the dog runs AROUND the squirrel, then yes, the dog runs around the squirrel no matter what the squirrel is doing. If, on the other hand, the dog is only running in a bigger circle around the tree than the squirrel is, but the dog never actually runs around the squirrel, then no.

Yes, it's just hoping to make the squirrel dizzy. Kidding.

Lol, it's still chasing after the squirrel, but it doesn't run around the tree... It sees where the squirrel is, and run near it, followed by jumping about, hoping the squirrel will fall from the tree. Hehe. Unless it learns to climb a tree, it's gonna still run around like a mental patient.

Why do you want to know? A dog chased you up a tree?

no, the dog is running around the tree that the squirrel is running around... but he is still chasing the squirrel.... haha...

It must be. In the statement you said it is so ("dog runs around a squirrel"), therefore it must be so.

Yes - think of the moon spinning around it's axis, while spinning around the earth, while the whole thing spins around the sun, all while being part of a spinning galaxy.