I believe the answer is different for East and West. For a traditional Western Definition I checked the Catholic Encyclopedia (newadvent.org). Here's what they say:
"Inasmuch as the relations, and they alone, are distinct realities in the Godhead, it follows that the Divine Persons are none other than these relations. The Father is the Divine Paternity, the Son the Divine Filiation, the Holy Spirit the Divine Procession. Here it must be borne in mind that the relations are not mere accidental determinations as these abstract terms might suggest. Whatever is in God must needs be subsistent. He is the Supreme Substance, transcending the divisions of the Aristotelean categories. Hence, at one and the same time He is both substance and relation. (How it is that there should be in God real relations, though it is altogether impossible that quantity or quality should be found in Him, is a question involving a discussion regarding the metaphysics of relations, which would be out of place in an article such as the present.)"
I think this is pretty far from tri-theism.