Plenty of schools

As usual with Japanese culture (in my opinion), there's also many differing schools on making Ikebana.

Ikenobo - The oldest school which is based on having "The Three Elements" within the Ikebana.

Rikka - Used exactly nine elements within the arrangement. Very formal and precise. If I recall correctly it was used by the "royalty" of the time and learning to make it was one of the norms back then.

Nageire - Relaxation and freedom as the core, this was once a must to learn for every "sophisticated" woman.

Shoka - Bringing together Ikenobo and Rikka, Shoka is precise but allows the maker more freedom over the material and it's use. The thought of "The Three Elements" shows slightly through with the rule of having three branches as a must in the arrangement. The branches symbol "Human, Sky and Earth"

Ohara (aka. Moribana) - Common traits for this is western heavy flowers and a shallow holder. I take it that as the western culture started to appeal more to the Japanese artists, the curiosity to use flowers from another region seemed interesting.

Sogetsu - The natural conclusion of every art form eventually in my opinion. Sogetsu is all about absolute freedom and the downfall of old rules and practices.