Son has dry deep cough a lot last 3 winters persistent had 1 allergic react? On last spring could it be ashtmatic no other symptoms how best to treat

It may be asthma. Persistent dry coughing may indicate constriction of the airways. There is a history of allergic reactions which place your son at higher risk for asthma. Your doctor can perform a lung function test with and without a bronchodilator to determine the response.
Cough varient. There are milder variations of "asthma" that may emerge at any age and show up only as cough. Wheezing may never occur. For these kids asthma meds are helpful in eliminating the cough while traditional cough meds may do nothing or worsten the sx. Your dr may be able to show subtile changes in pulmonary function tests. I've lived with cough varient asthma for decades while sis had a fullblown case.
Asthma is likely. Treatment depends on age and severity of symptoms. Options include inhalers, pills or nebulized medicines (liquid that is misted, then inhaled). Ask your physician about what is the best approach. If he has a rescue inhaler like proventil (albuterol) and needs it daily, then inhaled steroids daily are the treatment of choice. Best of luck!
Possible. But there are also other causes for dry cough with the most common ones being post-nasal drips (from allergic rhinitis) and sinus infection. He needs a lung function test to tell if he has asthma.