Cities will only cut down trees or trim trees that extend out into alley ways or into a public street that need to be cut down for safety or visibility reasons.
Usually though if the trees is on your property and is not causing any problems with the street or alley way then it's usually the homeowners responsibility to cut down the tree on their property.
Having a professional cut down a tree costs around $1,000.00 to $1,800.00 depending on the size of the tree and any other things such as how close it is to the house etc.