I've finished all four books and still do not understand the purpose of the Tree of Thorns from the perspective of any of the factions?
It's been a while since I read them, so I can't remember in which book this was explained:
The human UI (Empathy) had originally fled from its confrontation with the Machine UI by travelling back in time. The Tree of Thorns was designed by the Machine UI to serve as a beacon of agony which Empathy could not ignore, and would force it to come forward so the Machine UI could defeat it.
It is mentioned near the end of The Rise of Endymion that Aenea had managed to "control" or "harness" the Shrike, something like that. So its role in the latter two books doesn't necessarily jive with its larger mission.