There's an entire subreddit full of stories of bad neighbors cutting down trees that don't belong to them and the homeowners receiving punitive 5+ figure settlements.
How valuable is a tree? First, trees have value as lumber. A large tree (say, with a trunk of 20" diameter) can easily provide 500-600 board feet of lumber, which can be worth about US$1000 wholesale, and more than that at retail prices (the value depends on the quality and the species). Trees contribute to the value of land. The contribution of an individual tree is usually not very large; one common estimate is about 2-3% of the land value, for a typical suburban block, if the tree is large and attractive. This could easily be $5-10,000.
The protection of private property is firmly rooted in American law (and the law in many other countries), and it should be no surprise that the destruction of property can bring significant legal penalties, and can result in compensation paid to the property owner. For negligent destruction, the penalty might be limited to the damages caused. For a tree, this could be the value of the tree, or the cost of replacement (including removal of the old stump, etc.). For deliberate destruction of property, there can be a penalty of a sum of money (e.g., $10,000) or three times the value of the property (or 5 times, or 10 times, depending on the value of the property and the state).
Combining a penalty of triple damages with a value of the tree of about $5-10,000, and/or replacement costs of many thousands, easily results in 5-figure penalties and/or compensation for a large, but not extraordinary tree. In some cases, there can be additional compensation for aesthetic loss and mental anguish. The protection of trees simply results from the legal protection of property, and trees on private land being property, owned by the land-owner. Trees have value, and destruction of trees can be the destruction of valuable property.
For discussion of the value of trees, including size, species, and location (i.e., including impact on property value), see:
Note the 3 examples of trees on page 5, valued at $12,066, $3,670, and $39. Some trees are quite valuable (and would result in large penalties at triple damages), and others much less so.
For more on value as lumber, see https://www.extension.purdue.edu/extmedia/FNR/FNR-93-W.pdf