The same priority on property rights applies to trees. I can't cut down a tree on your property, but I can cut down a tree on my property. The town in the article made a assertion that is no weirder than corporations being considered "persons" with "rights", yet that is widely accepted in our society.
In fact, corporate "personhood" is even weirder: This town did not make a law to enforce trees rights. However, applying "personhood" status to corporations is written into law all over the place even though corporations are a human construct, not sentient beings. So, again, the only way the current laws are logical is to see that they are all about enforcing property rights, not out of concern for trees, animals, and -- at one time -- humans.