How Frieza’s ancestors chose their successors: The successor had to kill their entire family, sparing only themselves. However, their genes were deeply imprinted with the instinctive love for family—it was literally written into their genetic code. Therefore, only the strongest, most ruthless individual who could overcome their own genetic instincts could become the next ruler. Even if the successor was unwilling, their father would force them to carry out the killings, including killing the father himself.
However, King Cold put an end to this brutal tradition.
To safeguard the empire's future, the creator of this tradition also embedded a countermeasure into their bloodline: a genetic rule that prohibited parents from killing their own children. This was designed to prevent any ruler from destroying their offspring out of fear of being overthrown, ensuring that power struggles wouldn’t stifle the empire’s development and innovation.
There are no side branches or distant relatives in the Frieza family—because the founder of that tradition wiped them all out. (This cruel system wasn’t created by the very first emperor, but was established many generations after the original founder.)