I know of no tree that could such a thing.
Tree roots going down go down via new growth.
Once this new growth is there it stays there a lifetime.
Only disturbances may cause downward replacement.
Some bulbs however do manage to grow downward a little distace.
So what kind of tree would do such a thing?
I would suggest the can got there when planting or uprooting the tree, but I suppose you already considered that and discarded it for some reason.
Do what kin
Thanks for plausibly identifying Beth Togarmah with (the heartlans of) Armenia.
It should be no surprise at all that the Catholic church is not joining the search.
They have written record it landed somewhere else.
(And besides that, when was Turkey allowing foreigners, who have good contacts with Armenian and Russian orthodox, to enter and survey near it's eastern border?)
Both the Septuagint and the Masoretic text state in Genesis 11:2, which is just after the story of Noah's ark and the genealogy of his offspring, that the people went westward and settled in the plain of Shinar.
Septuagint, Douay Rheims translation:
"And when they removed from the east, they found a plain in the land of Sennaar, and dwelt in it." (http://drbo.org/chapter/01011.htm)
Masoretic, King James translation:
"And it came to pass, as they journeyed from the east, that they found a plain in the land of Shinar; and they dwelt there." (https://biblehub.com/kjv/genesis/11.htm)