If you only have New York's minimum required auto insurance then no, this would not cover damages to your own vehicle.

New York state law requires minimum Bodily Injury Liability limits of $25,000 per injured person up to a total of $50,000 per accident, and Property Damage Liability coverage with a minimum limit of $10,000. This basic coverage is often referred to as 25/50/10 coverage with 25/50 UM limits.

The state of New York follows a No-Fault system meaning your insurance company will make payments for your injury claims regardless of fault, up to a specified limit. Personal Injury Protection (PIP) in your coverage helps pay for "reasonable and necessary" medical expenses for you and your passengers. New York state law requires a minimum Personal Injury Protection coverage of $50,000.

To help protect against damages caused to you and your passengers by an uninsured driver, New York state law requires a minimum Uninsured/Underinsured Motorist coverage of $25,000 per person up to a total of $50,000 per accident for any bodily injury caused by the uninsured and/or underinsured driver.

So in NYS the following insurance is required:

Bodily Injury Liability: $25,000/$50,000 Limit
Property Damage Liability: $10,000 Limit
Uninsured/Underinsured Motorist Bodily Injury: $25,000/$50,000 Limit
Personal Injury Protection: $50,000 Limit

So if you were injured in your accident when your vehicle collided with the tree due to the snowy weather you should be able to make a claim against your PIP coverages. You do no however have coverage for your vehicle without having the extra physical damage coverages of collision and comprehensive.

In your situation where you lost control of the vehicle in the snow and hit the tree you would have needed collision coverage in order to make a claim for the damages to your car. This is why it is important to keep "full coverage" on your car, meaning state required liability coverages plus collision and comprehensive, even if you have paid off your lien holder or paid for your car in full when you purchased it. Without physical damage coverages if you are in an accident where you are at fault you will have to pay for the repairs yourself or be out of a car if the car is a total loss and cannot be repaired.