Generally, we instead have animal welfare laws that protect various animals to various extents for various reasons, based on human interest in said animals (e.g. You can sterilize any cat you find, unless it's owned by someone else, but you can never shoot a cat; you can shoot many wild animals within certain limits, but you can't sterilize them outside very special circumstances).
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.
The general point is that animal protection are almost entirely subsumed to human rights - animals are protected in so far as their protection helps humans in some way (either specifically, such as your chicken being useful to you so that no one else can kill them; or environmentally, such as elk being important for the health of certain forests). Given the human need to consume or displace other living beings, this is the only tenable moral position that can be held anyway.
This seems roughly in line with how we treat certain wild animal populations though.