11 December 2009

How many descendents? How many cousins?

Following up on my recent posting on counting the cousins ... how many times have you read articles and blog postings that deal with how many ancestors you have? Two parents, four grandparents, eight great-grandparents, and so on. The factor two growth generation by generation is powerful, to the point that after several hundred years you get to more ancestors than the population of the earth. The theoretical dilemma is resolved not by speculation on alien invasion but by cousin marriages that drastically reduce the actual number of ancestors.

I don't recall seeing the problem approached from the other way around, looking at descendants. It's more tricky because while you know each child has two parents you don't know a priori how many children they have.

Let's play a bit. Here's a scenario, a story. Imagine a highly disciplined village populated by six couples. Nobody leaves or moves into the community.


Mr. and Mrs. Zero have no children so no further descendants. Their graves in the old churchyard lie neglected.

Mr. and Mrs. One have one child and each succeeding generation has one child. In any generation there is just one child. The family history is treasured like a string of pearls. Graves in the old family plot are carefully tended.

Mr. and Mrs. Two have two children. Their children each have two children making four grandchildren, and so on. Eight great-grandchildren in the third generation is two raised to the power three. The original Mr. and Mrs. Two have 1024 descendants in the tenth generation, two raised to the power ten. Generally for the nth generation it's two raised to the power n.

It's possible, and even likely, that in the intervening generations cousins will have married meaning that 1024 descendants in the tenth generation is the maximum.

A whole section in the old churchyard is devoted to the Two family, but it couldn't accommodate them all.

Mr. and Mrs. Three delight in their three children, nine grandchildren, and 27 great-grandchildren. By the tenth generation there are a maximum of 59,049 descendants, three raised to the power ten.

As time goes on there's a building boom in the village to accommodate the fecundity of the larger families with a new larger cemetery opened.

Mr. and Mrs. Four enjoy four children, 16 grandchildren, 64 great-grandchildren, and by the tenth generation the Four family descendant count is 1,048,576 members, four to the power 10. They are dwarfed by the 9,765,625 tenth generation descendants of Mr. and Mrs. Five.

The new cemetery becomes many cemeteries and they are overwhelmed. Cremation becomes fashionable.


Now let's look at the cousins. People in the tenth generation are ninth cousins unless more closely related by being descended from a more recent ancestor. For a given Two family descendant half are ninth cousins, half are more closely related. That's assuming no cousin marriages.

Amongst tenth generation descendants of the Three family two-thirds are ninth cousins, one-third more closely related. For the Four family three-quarters are ninth cousins, and for the Five family four-fifths. That makes 7,812,500 ninth cousin Five family descendants in the city which was the village.

Uneven family size

The situations get more complex to analyze when family size varies from sibling to sibling and generation to generation.

Let's look again at the descendants of Mr. and Mrs. Two, but imagine that one of their two children had a rebellious streak and instead of the allotted two children had 12. With that the rogue gene was played out and each subsequent generation stuck to its two offspring. By the 10th generation the original family progenitors would have 6,144 descendants of their rebellious child, and 512 from their other child. The 6,144 would have 512 ninth cousins, and the 512 would enjoy 6,144 ninth cousins.

Obviously family size is hugely important, but not as important as these figures would suggest. Although I've yet to identify any examples in my own family tree, marriage between more distant cousins is, must be, remarkably common.

Take a look at your family tree. Are there any siblings a few generations back who had particularly large families? Those are where you might be most likely to find connections to distant cousins, perhaps through the emerging technology of autosomal DNA analysis.


DWP said...

Some useful and some not-so-useful discussion related to this subject can be found by searching the Internet for "pedigree collapse"

gordon said...

My late wife was born in lac St-Jean area in the 1930s when large families were still in vogue. As with many Quebecers I have been able to fill most of her pie chart to the 10th generation except for a the semi-octant for her Irish gt grandmother who was orphaned on the way over in 1847. There are a fair number of repititions as one goes around the chart in eighth (and earlier) generations. I would think most pure-laine Quebecers would be seventh or eighth cousins. My wife has some 65 first cousins!