Who has the duty to remove a tree causing an obstruction on the highway?

Negligence is the failure to exercise due care (that degree of care in this case that a reasonable state highway authority would have exercised  under the same or similar circumstances to prevent foreseeable harm).  There is also negligence for failure to exercise due care (that degree of care in this case that a reasonable driver would have exercised under the same or similar circumstances to prevent foreseeable harm).

Some states have comparative negligence where both parties are liable.  If the state has comparative negligence, a percentage of fault is determined for each party.  Although the state highway association was negligent for not having removed the fallen tree, you were also negligent for talking on your phone and not seeing the tree, not wearing a seatbelt and having had a couple of drinks before driving.

A percentage of fault would be determined for both you and the highway association. The following figures are only for purposes of an example and are not the actual figures.  If you are held to be 60% negligent and the state highway association is held to be 40% negligent, this means that you could only recover 40% of your damages (property damage and personal injury) from the highway association.  Your property damage claim is separate from your personal injury claim.  You can file a claim with the state.  Don't miss any filing deadlines when filing a claim with a state agency because missing the deadline to file an administrative claim may result in your right to file a lawsuit upon denial of the claim being denied for missing a filing deadline.  It may be possible to settle the case with the state agency's insurance carrier.  Your damages would be the property damage to your car and your separate personal injury claim.  If the state agency is accepting liability, you could get medical treatment and when released by the doctor upon completion of your medical treatment, file your personal injury claim which would consist of your medical bills, medical reports and documentation of any wage loss.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of your injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  If you are dissatisfied with settlement offers from the highway authority's insurance carrier, reject the settlement offers and file your lawsuit for negligence against the highway authority.  You should also name the state as a defendant in your lawsuit.  If the case is settled with the insurance carrier, NO lawsuit is filed.  If the case is NOT settled with the insurance carrier, you will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.