How do you know if a tree removal company is actually reputable?

We have a pretty tree in our yard in New Jersey that might need to come down after the last storm. I started looking into tree removal companies and honestly it is a little confusing.

Some quotes online seen really cheap and others are extremely high. I have also heard stories about companies damaging properties or not being properly insured.

For people who have hired tree services before, what should you actually check before hiring someone? Are there certain red flags to watch out for or certifications that matter?