Removing a tree jointly owned with the city

It's a distinctive tree, clearly ill and with some branches looking rotted out. I've had a distinctive tree removed that was entirely on my property, so I'm aware of the permit process for that.

Is there any chance the city would pay (even partially) for removal of a sick/dangerous tree, given the joint ownership?

If I'm financially responsible for removal, does joint ownership change the process at all or would I just hire someone of my choosing and apply for the distinctive tree permit as if it was fully on my property?

Thanks!