I want to ask whether a certain abstract structure has to have children of the same kind, possibly nested an infinite number of times. The closest example would be a genealogy tree where each node has scalar properties, like first name or gender, and also successors, each having scalar properties and successors, and so on. So, should I ask, "should this structure be hierarchic" or "should this structure be hierarchical"? What's the difference between the two?
Your question seems to revolve around a computer science data structure problem of whether nested objects are hierarchical.
They are not.
However, they can sometimes be transformed into a tree-like structure and be hierarchical.
You should use hierarchical since it is part of the technical terminology for computers.
hierarchical data structures
hierarchical hidden Markov models
hierarchical k-means tree
The only phrase I've come across using hierarchic is hierarchic escalation which means pushing things up the chain of command.