Hi and welcome to the forum! I am glad that I can help you.
You would like to know the cause of breathing problems after running.
Individuals may experience shortness of breath or difficulties breathing due to acute contraction of the bronchioles. This condition is called exercise- induced asthma. Individuals may not experience allergies when experiencing this condition. People who when young developed frequent lung infections such as bronchitis may develop asthma that is not allergy induced.
Given the symptoms that you described, you may or may not experience exercise induced asthma. You may consider seeing a pulmologist(specialist for lung diseases) to rule out asthma. You may also consider seeing a cardiologist to rule out any other underlying heart condition.
Please keep in mind that I provide medical information only. I am not able to diagnose medical conditions online. Please contact your doctor or other qualified healthcare provider such as a pulmologist and a cardiologist for further advice and information about diagnosis and treatment options of your current condition.