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Will homeowners insurance pay to remove a hazardous tree?

I have a tree near the house that should be removed before it crushes the house. Seems that it would be smart for insurance to cover this rather than pay for repairs later.

Your insurance company will not cover the tree removal as it stands. Your policy reads that you are expected to take preventative measures with any hazard that may cause a loss. If you know about it and don't remove it before it causes a loss, your policy may exclude coverage.

Your insurer would have sent you a policy form (it looks like a book) which will define anything and everything you need to know. Just about every insurer follows the same ISO policy form which indicates they will pay $1000 for the removal on the residence premises if (1) your tree fell due to the peril of windstorm, hail, weight of ice, snow, or sleet (2) a neighbor's tree fell due to a covered peril provided that fallen tree damaged a covered structure, blocks a driveway on the residence premises or blocks a ramp which is designed for use by handicapped persons entering or leaving the premises.

It would be smarter for you to cut down the tree. They may pay for the repairs, but you live with the inconvenience. Stop being cheap.
Most likely, they would. BUT, the cost to remove a tree is usually less than the deductible you would have to pay, before the insurance company pays anything.
The simplest way to find out is to call your agent. The answer is free.
Not a chance. And if the tree does fall, homeowner's deducible will probably be more than the cost for you to cut it down yourself, not to mention your insurance will probably go up after the loss.
Call your agent. Also if the tree is near any power lines the electric company might pay to remove it.
It will not pay for the tree to be removed. It will pay for the damages it causes when it falls.