pulmonologist, i got diagnosed with serious asthma problems since i was 5 months old. ive been going to a pulmonologist ever since twice a year. And i found out i would probably have it my whole life. So i would recommend. just a pulmonologist i dont think a pediatric is necessary.
Both the asthma specialist and the pediatric pulmonologist are both capable of handling your daughter's condition. The only difference is that the pediatric pulmonologist is more experienced with children's respiratory problems.