D&T Gill hyperplasia
Gill hyperplasia is essentially the build-up of cells in a particular area, in this case being the gills, causing the gill cover to swell and open.
Gills are extremely sensitive to toxins and issues such as ammonia, nitrite, high amounts of nitrate, some medications, chlorine and chloromines, wounds caused by other fish or tank decor, and illnesses causing irritation (such as parasites and flukes). As gill tissue does not tend to heal extremely well, it can tend to grow another layer of cells causing the gill cover to open (hyperplasia).
Generally, exposure to toxins would have already occupied prior to purchasing the betta, so improved water quality will always help issues. If the hyperplasia is quite severe, it is usually irreversible. Despite some extremities of it not being reversible, there are methods you can take to prevent further damage or issues such as maintaining good water quality, and adding a small amount of dissolved aquarium salt to the tank (0.5tsp per 5 gallons) to help improve gill function. Hyperplasia in bettas will cause the betta to rely on the labyrinth organ a lot more, so having water of little depth or surface dwelling plants will aid in making the bettas life a lot easier.
Sadly, in some cases, hyperplasia will also affect the labyrinth, without the labyrinth organ, bettas cannot survive as they will be starved of oxygen. Extremities such as labyrinth damage usually only occur in very poor water conditions or long exposure to ailments such as parasites without treatment.
Methods to prevent this issue also involve maintained water quality, use of soft materials within the tanks (silk or live plants, and not plastic ones) , as well as non-aggressive tank mates if your betta will live happily alongside other fish.
This image, although not a real hyperplasia case is an example of the gill plates being pushed up on either one or both sides permanently.
Hyperplasia is not very common at all with bettas kept in good conditions (even in poor conditions for that matter), however is another form ailment-causing growths that is most common in bettas exposed to poor water and housing conditions.