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The branch theory is a theological hypothesis within Anglicanism, holding that the Roman Catholic Church, the Eastern Orthodox Church and the Anglican Communion are the three principal branches of the One, Holy, Catholic, and Apostolic Church. Some Anglican theologians also include the Oriental Orthodox churches, the Assyrian Church of the East, the Old Catholic Church, and the Church of Sweden.
The Oxford Dictionary of the Christian Church defines the branch theory as:
…the theory that, though the Church may have fallen into schism within itself and its several provinces or groups of provinces be out of communion with each other, each may yet be a branch of the one Church of Christ, provided that it continues to hold the faith of the original undivided Church and to maintain the Apostolic Succession of its bishops. Such, it is contended by many Anglican theologians, is the condition of the Church at the present time, there being now three main branches…
William Palmer (1803–1885), an Oxford theologian, was the principal originator of the Branch Theory. His two-volume Treatise on the Church of Christ (1838) formulated the notion. The theory was then popularized during the Oxford Movement, particularly through the work of the Tractarians. However, some leaders of the movement became dissatisfied and later became Roman Catholics.
Although the Anglican Roman Catholic International Commission, an organization sponsored by the Anglican Consultative Council and the Pontifical Council for Promoting Christian Unity, seeks to make ecumenical progress between the Roman Catholic Church and the Anglican Communion, it has made no statement on the topic, and no support for the branch theory has been expressed anywhere outside Anglicanism itself.
Roman Catholic 
The Roman Catholic Church holds that maintaining the teachings of the ancient Christian church is essential and that there are other churches in which there is valid apostolic succession with valid orders, even if those churches are in a state of schism or heresy. But it does not accept that those separated by schism or heresy are part of the one church. Instead, it teaches that "there exists a single Church of Christ, which subsists in the Catholic Church, governed by the Successor of Peter and by the Bishops in communion with him". It considers Anglican orders invalid in general and holds that, though individual Anglicans may have orthodox faith, the Anglican churches have not maintained the fulness of ancient Christian teachings, most notably on the sacraments.
Eastern Orthodox 
The Eastern Orthodox believe that the one true church founded by Jesus Christ is a real identifiable entity and that it is singularly the Eastern Orthodox Church. This church has identified itself as the "one, holy, catholic, and apostolic church" in, for instance, synods held in 1836 and 1838 and in its correspondence with Pope Pius IX and Pope Leo XIII. Adrian Fortescue wrote of the Eastern Orthodox: "The idea of a church made up of mutually excommunicate bodies that teach different articles of faith and yet altogether form one Church is as inconceivable to them as it is to us (Catholics)".
Those who attack the Church of Christ by teaching that Christ's Church is divided into so-called "branches" which differ in doctrine and way of life, or that the Church does not exist visibly, but will be formed in the future when all "branches" or sects or denominations, and even religions will be united into one body; and who do not distinguish the priesthood and mysteries of the Church from those of the heretics, but say that the baptism and eucharist of heretics is effectual for salvation; therefore, to those who knowingly have communion with these aforementioned heretics or who advocate, disseminate, or defend their new heresy of Ecumenism under the pretext of brotherly love or the supposed unification of separated Christians, Anathema!
- See The Christian Faith: An Introduction to Dogmatic Theology, by Claude Beaufort Moss, SPCK, 1943, p. 279, available online at http://orthodoxanglican.net/downloads/faith.pdf
- The Oxford Dictionary of the Christian Church (Oxford University Press 2005 ISBN 978-0-19-280290-3, article "branch theory of the Church"
- Dominus Iesus, 17
- Apostolicae Curae of 1896
- Michael Plekon (editor), Tradition Alive: On the Church and the Christian Life in Our Time (Rowman & Littlefield 2003 ISBN 0-7425-3162-7), pp. 230, 237
- Dissertations on Subjects Relating to the Orthodox or Eastern-Catholic Communion quoted in Anthony Dragani, Adrian Fortescue and the Eastern Christian Churches, p. 81
- Erwin Fahlbusch, William Bromiley (editors), The Encyclopedia of Christianity (Eerdmans 2003) vol.3, p. 867
- Adrian Fortescue, The Orthodox Eastern Church, p. 366
- The ROCOR's Anathema Against Ecumenism (1983) (Orthodox Christian Information Center)