The Sanlun school, also known as the Three Treatise school, is a Chinese Madhyamaka school that emerged during the Tang dynasty in China. It is based on the teachings of Nagarjuna and his commentaries on the Madhyamaka philosophy.

The Sanlun school is divided into two main branches: the Northern branch and the Southern branch. The Northern branch was founded by Jizang (549-623 CE) and the Southern branch was founded by Kumarajiva (344-413 CE).

The Northern branch of the Sanlun school is also known as the "school of emptiness" because it focuses on the concept of emptiness (śūnyatā) as the ultimate reality. Jizang's teachings emphasized the importance of understanding the emptiness of all phenomena in order to attain enlightenment. He also emphasized the use of logic and reasoning to understand the concept of emptiness.

The Southern branch of the Sanlun school, founded by Kumarajiva, is known as the "school of consciousness." This branch focuses on the concept of consciousness (vijñāna) as the ultimate reality. Kumarajiva's teachings emphasized the importance of understanding the nature of consciousness and how it relates to the concept of emptiness.

Both branches of the Sanlun school share the same core teachings, but they differ in their approach and emphasis. The Northern branch focuses more on emptiness, while the Southern branch focuses more on consciousness. However, both branches agree that the ultimate reality is beyond conceptualization and can only be understood through direct experience.

Apart from these two main branches, the Sanlun school also has sub-schools that emerged later on. One of these sub-schools is the "school of middle way" founded by Fazang (643-712 CE). This sub-school emphasized the concept of the "middle way" and the idea that all phenomena are empty of inherent existence.

Another sub-school is the "school of perfect enlightenment" founded by Zhiyi (538-597 CE). This sub-school emphasized the importance of meditation and the practice of the Six Perfections (pāramitās) in attaining perfect enlightenment.

In conclusion, the Sanlun school has two main branches, the Northern and Southern branch, which differ in their emphasis but share the same core teachings. It also has sub-schools that emerged later on, each with their own unique approach to understanding the concept of emptiness and attaining enlightenment.