The Branch Theory is a belief, outside of Orthodoxy, that communities of Christians that are not in communion with each other, may still be a branch of the One Church. Recently this has been expressed as a two lung theory by Roman Catholic writers, or there being three main branches by Anglican writers. Sometimes, by others, the entire “tree” is call an invisible church.
Visible unity of faith
As a matter of faith, Orthodoxy believes in the bond between God and his Church, the Church is one, even as God is one. There is only one Christ, and so there can be only one Body of Christ. Orthodox theology does not separate the invisible and the visible Church, and therefore it cannot say that the Church is invisibly one but visibly divided.
Unity is one of the essential characteristics of the Church, and since the Church retains its essential characteristics, it remains and always will remain visibly one. Orthodox writers point out that there can be schisms from the Church, but no schisms within the Church. Schisms do hurt the Church but they cannot affect the essential nature of the Church.
Branch Theory implies that many, if not all, Christians communities can trace their history, through other “branches of Christianity” back to the Apostolic Church. The Orthodox Church believes that more is necessary to claim apostolic succession. The community needs to hold the apostolic teachings, and be as one with all who hold the faith. To choose to hold other beliefs is heresy.
- Orthodox Christianity and The "Branch Theory" - OCA Questions and Answers About Orthodoxy