Throne inheritance without an initial king

Model a royal family's line of succession: people are born to parents, people die, and you return the current order of inheritance (a pre-order walk of the family tree that skips the dead). This is the classic throne-inheritance design, with a twist: the constructor does not give you a king, so the structure must start empty and decide who the root is as people are added.

Explain how the twist changes your data structures and edge cases.