I guess one could look at it as the church was one until it split and so in that way the Roman Catholic Church is a branch as much as Eastern Orthodoxy. In an organizational sense this would seem to be fairly accurate.
From a theological sense all branches are claiming to be the true church so the branching graphic would not make sense there as they consider themselves to be true and their heritage to be corrupted. The branch idea does not really make sense as the assumption is your root is diseased. Each church would consider itself a new shoot off of the original church.
The Eastern and Latin Church indeed seem to be one. I believe it was Pope John Paul II who referred to the Eastern Church and the Roman Church as the two lungs of the same body. But isn’'t that the Eastern Catholics, (i.e. Coptic, Byzantine, etc.?). Either way, I expect that the Greek Church, the Russian Church etc. will one day all acknowledge the full deposit of faith to be in Rome, and come to accept the primacy of the Apostle St. Peter, as well as all the councils. If the EO and the Roman Church are collectively the One Holy Catholic and Apostolic Church, then it will be so. I don’t feel anywhere nearly qualified to take on the issues of Eastern Orthodoxy and Roman Catholic Orthodoxy. They both have fully maintained apostolic succession, and maintain a valid episcopacy and priesthood who may validly confect the Holy Eucharist, and stand in persona Christi. I’ll save the split of the Catholic Church for a time of my further education, and for other threads. But protestantism still doesn’t hold up in any way that I am able to understand with the one exception that we share a deep and mutual understanding, (for the most part), of who Jesus is, what He did/does, and why He does/did it. This is no small thing, and I am filled with deep love of my protestant brothers and sisters. But all the same, they are clearly, and unquestionably branches, sects, and denominations by their very definition, and the Catholic Church (Eastern and Latin), clearly is not any those things, having been established by Jesus Christ our Lord in a direct and historical way. This may very well apply to the Eastern Orthodox as well, though I am admitting to not possessing enough knowledge even to speak intelligently on that matter. It is my next topic of deep study in Church history. Martin Luther, John Calvin, et. al. fully broke with the Church with a completely different understanding of some of the most fundamental ideas of the application of Christianity on the life of man. Dropping the oh, so important concepts of magisterium, infalibility, tradition beside scripture, understanding of the episcopacy, the sacraments and more. Each of these groups then in turn STILL breaks up further and further based on internal group politics, and many are even flexible and comport to the cultural ups, downs and mores in which they find themselves, so that they, in many cases, no longer even adhere to the principles and ideas of their own founders, who were not Jesus or the apostles to begin with. The mega church down the block from me bears very little, if any, resemblance to Church established on St. Peter, or even on the amended versions of church promoted by Martin Luther and John Calvin. It seems more of a meeting of people to hear a motivational speaker who is friendly too many of the teachings of Jesus Christ, as each member may interpret Him to be.
Go into most any Roman Catholic, Eastern Catholic, (and probably Eastern Orthodox) Church, and you will find them conducting Christian worship, and sacramental life pretty much they way it was conducted in the 1st century. (see the Didache, or read any writings of the early Church fathers).
Having said all that, the comments here have helped me to understand a little bit, what it is the protestants believe about their own groups. I couldn’t even get my mind around THAT before.
As to the main question of this thread, I think the answer is most likely, never. But here again, it goes back to what does one MEAN by “legitimate”.
Thanks to all for your explanations,
May God bless,
Steven