Wednesday, December 16, 2015

The Mathematics of Genealogy

When most people begin looking into their genealogy, I tend to believe their thinking is much like mine was when I first started out, which is something along the lines of this, "I have two parents and two grandparents. I see this branch of my family tree goes back 5 generations so I have 10 ancestors!"

In reality, the growth in the number of ancestors is exponential. Every generation you go back, you double the number of ancestors you have. This make sense on the surface, but again, I don't believe most people fully grasp what this means. If I was able to trace back every single branch of my tree as far back as I have my Brandstetter line, the numbers are virtually astronomical.

Christian Brandstetter and his wife, Maria, are my 8x-great-grandparents. This means they are ten generations back, not including myself. To find the actual number of ancestors at that level of ancestry, you need to double your number of parents (2) by two, ten times. This does not result in 10 as some are likely to do. Take 2 and double to 4. Then double 4 to 8, then double 8 to 16. At this point, you are only back to your great-great-grandparents. All 16 of them. Now you need to double this 16 another six times! 16 x 2 = 32 x 2 = 64 x 2 = 128 x 2 = 256 x 2 = 512 x 2 = 1024. At the level of grandparent that Christian Brandstetter is to me, I have a total of 1024 ancestors! Over one thousand!

If you want an actual count of all ancestors at that point, you need to add together each of these subtotals. So, to count all of my ancestors at all levels going back to Christian's generation, you need to add 1024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 to come up with the total of 2046 ancestors. To put history into perspective, this only takes me back to around the year 1600. Keep on doubling all the way back to the time of Christ and how many do you have? It boggles the mind.

Now, keep in mind there is a thing called generational collapse. This means that some of your ancestors were actually related to each other. No, this doesn't (necessarily) mean you had brothers and sisters as parents to one of your ancestors. It means that there were 10th, 7th, 5th, second and even first cousins that married and had a child who ended up being one of your ancestors. It can also mean that a pair of your ancestors were very distantly related. So distant, in fact, that they shared little, if any, actual DNA.

Now, turn this around the other direction. How many descendants did your ancestors have? I can almost guarantee that most generations had more children than recent generations have had. Yes, there was more infant mortality and childhood diseases but still, they typically had many children that grew to adulthood and had children. I honestly don't know what the average number of children previous generations may have had that had children of their own. Just for the sake of an example, let's say there was an average of three that produced offspring in each generation. I tend to believe this is on the low side but it will work for this purpose.

If 1024 people (512 couples) each produced 3 children, that's already over 1500 children in the first generation! Then 4500 in the second and nearly 15,000 in the third generation. Now, you also need to keep generational collapse in mind. Some of these will have had children with each other but the number that did, I believe, were relatively small, especially in later generations, once people started moving around the world over the past 200 or so years. Even considering all that, I am likely related to thousands of people who are descended from at least one of those 512 couples 400 years ago.

As my mother likes to say, "You go back far enough, we're all related."

And people wonder why you can't focus on one branch of the family tree after you get your research rolling.

 One more bit of mathematics. This post marks my one hundredth blog post. Seriously? 100 blog posts about genealogy? It has taken almost three years to post them. I never thought I'd make it but the time has flown by.


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