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How Y chromosome and mtdna work
There is some confusion about how DNA works. The chart below shows what the two types used in tracing backgrounds can show. I used a male because males receive both the Y chromosome DNA from their fathers and the mtDNA from their mothers while females do not receive the Y chromosome. (However, males do not pass their mother's mtDNA on to the next generation; it stops with that male and his children get THEIR mother's mtNDA..
With this kind of tracing, you can get only ONE line in the male and ONE line in the female. There is no way to cross over from male to female lines. The DNA goes straight down one line or the other. Only these types of DNA remain pure; that's why they are the only ones used in genealogical work.
The arrows show who each person received the Y or mtdna from.

This chart shows that you have a lot of ancestors who will NOT show up with either type of test. You will receive genetic material from all your ancestors defining eye color, complexion, and other features, but these genes combine and re-combine and there is no way to tell where it came from. Y chromosome DNA and mtDNA do not combine, so they can be used to test ancestry. You can see why mtDNA cannot be used to trace a surname, as the surname changes each generation. This illustration is from my wife's German Schnake family.
Toenies Heinrich Brink or Schnake married Anna Maria Engel Schnake, the heiress to the farmstead, and changed his name to Schnake. This was common in old Germany. But notice--he was born Brink, so the male line would from this point on follow the Brink line. There would be no way to use DNA to follow the Schnake line any further, since there is no way to get Anna Maria Engel Schnake's father's DNA. He had no male heirs, which is why she inherited the farmstead.
If William Herman Schnake and another descendent of Toenies Heinrich Brink/Schnake submitted Y DNA tests, they should show close consistency. However, the test would not show anything about any of the other ancestors. Also, it could not be used to trace Schnakes before Toenies, because the name passed through Anna Maria Engel Schnake, a female.
William Herman Schnake is my wife's grandfather. There are no living persons on this chart. My wife could not use her DNA to trace anyone in this line. Her mother was a Kelly and that's where her mtDNA would come from. This is why mtDNA is not used much in genealogical research.