Shalom R' Daniel,

Great to hear from you!

The Halachic principle of "Bereirah," to which our Gemara is referring, means the following. Suppose a event occurs but with no determined designation. Suppose, however, that its outcome in the future will eventually have a definite designation. The principle of Bereirah says that the identity of that future outcome retroactively determines the designation of an event which preceded it.

A simple example would be the following. Suppose that before Shabbos a man placed two Eruvei Techumin, one to the east and the other to the west. And in doing so, he said, "If my rabbi will come to the east then the first is be my true Eruv, but if my rabbi will come to the west then the second should be my true Eruv." Bereirah says that the actual side, to which the man's rabbi eventually in fact will come, determines which Eruv was indeed the true Eruv. (See Eruvin 36b for more information about this example.)

So, you see, Bereirah means that that an eventual outcome retroactively determines -- or clarifies, hence the term "Bereirah" -- the designation of a preceding occurrence.

Only someone who says Yesh Bereirah, however, relies on this principle. Someone who says Ein Bereirah, on the other hand, disagrees and instead maintains that since the event in the past was undetermined when it occurred back then in real time, therefore it remains undetermined, even after the outcome has turned up one way or the other.

In our case, then, if you say "Yesh Bereirah", like Rashbag, then you are saying that the eventual side of the border, on which the fruit ends up growing, retroactively determines whether that fruit is associated with Chutz l'Aretz and therefore is Chulin or is associated with Eretz Yisrael and therefore is Tevel.

If, on the other hand, you say "Ein Bereirah," like Rebbe, then you are saying that the nourishment from the ground on both sides of the border -- Eretz Yisrael and Chutz l'Aretz -- gets mixed inside the tree. Eventually it is delivered equally to the fruits growing on both sides of the border, which is what results in each fruit ultimately taking on the status of being a mixture of Chulin and Tevel.

May you continue to attain greatness in Torah and Yiras Shamayim!

Warmly,

Yishai Rasowsky