Please don’t shoot the messenger… but it could be like this:
The street is likely on what’s called “right-of-way.” That is, the City has the right to use it for public transportation and utilities, and so you can’t build on it at all, and the City has the right to pave it with asphalt and concrete sidewalk, and every citizen has the right to walk on it and drive on it and everything else. But the City doesn’t own it as such. So the City can say that it’s your tree and mean it.
It gets a little worse, even. If you look at the map that got filed when your tract was subdivided (there will probably be one, but not absolutely always), the tract was possibly created with these streets already outlined. Even though the City doesn’t technically own the streets, there’s nothing on the map to show who does. Doesn’t make much practical difference, except in limited circumstances, because it would mostly only make much difference if the City abandoned (“vacated”) the right-of-way, which it probably will never do. But the actual ownership of the “parkway” will be vague, except that it’s right in front of your house.
It gets a little worse. Because the tree is private property, the City will probably not be interested in trimming it - except… the City will probably want to trim it occasionally to keep it from being a traffic hazard, but such trimming will happen when the City’s budget allows it to happen.
It gets a little worse. The City can tell citizens what they can’t build on their lots, often called zoning codes, and can also put some landscaping requirements in those codes, such as requiring that each lot has a “street tree.” So even though the City doesn’t own it, the City has some administrative power to say what happens to it.
It gets a little better, but not much. The City probably doesn’t care so much what happens to any individual tree, so long as it gets replaced - well, presuming that this isn’t a tree with any interest group surrounding it. If you call the local Public Works Department, they may have a brochure telling you what to do if you need to remove a street tree (um, probably at your cost). Um, the City probably doesn’t want wannabe lumberjacks to drop trees into the street, so it may have requirements for contractors doing the work.
The City may be politically interested in appearing green, and may be interested in urban reforestation or some such. That is, they may have some program to make it easier for you to replace the tree you cut down.
Regardless, you could try calling the local councilman or alderman’s office if you think that the Public Works Department is being a little harsh. Might not do much good, but probably couldn’t hurt.
Or maybe not much of this applies where you live. Er.