In much of the United States (and I assume Europe), there are a variety of environmental regulations which restrict what you can do with your land. I don't know them in detail, but I know that tree removal, wetland removal, changing watersheds, drilling/mining, storing toxic chemicals, etc. often require permits. This is the price of environmental regulations.
In Ireland, it is illegal to cut down any tree in the countryside if it's more than 30 meters from a dwelling and more than 5 years old, with a few exceptions.
Arguably, this is to protect what little woodland we have left. Practically, though, it means that I would _love_ to reforest my land with native species, but don't, because it will forever lose its value as grazing land.
And I tend to agree, though private property is a complicated thing. After all, who did you buy it from? And who did they buy it from? And the person before that? Unless you're royalty and can defend your property by force (arguably the only true allodial title), you're basically borrowing the land from whatever government defends it for you.
If the tree is endangered, or the root system is what is holding soil in place preventing landslides, or there is an endangered something else on the land you are sometimes blocked from cutting a tree.
California is famous for such regulations. In other states you are unlikely to have problems.
Sure, we're talking about the exceptions vs the rule. The rule, generally, is you can do whatever you want with trees on your private property. There are relatively rare exceptions in cases like you mentioned if an endangered bird has a nest on your tree or something.
