If you go back 12 generations you have 2^12 = 4 096 ancestors. Seems rather unlikely that every single one of them is Irish. If you go back 20 generations you'd have 2^20 = 1 048 576 ancestors, which is probably larger then the population of Ireland by that time.
While it's true there's likely an immigrant in the bunch, it's also very unlikely that all 4096 or 1048576 ancestors are distinct. The tree is bound to overlap.
The farther back you go the less likely it is that they would be immigrants. Back in the day when people had to walk or ride to where they wanted to go...
