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I was thinking about something the other day (that is probably stupid), and thought this would be a good place to talk about it.

If someone wanted to draw its family tree over the last 1000 years, he'd write a binary tree with a depth of about 40 (counting 25 years on average between each generations). This tree would have 240 leaves, or about 1012 ancestors on the deeper level. That is obviously much more than the total world population and a stupid figure when you think about it.

The only explanation that I can think of is that a real genealogy tree must have "linked" branches at several levels that change the actual number of ancestors. Anyway, for an average person, how many ancestors would that make? Could one assume that all or nearly all of the persons that lived in its country or world region a 1000 years ago would be part of its ancestors?

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Not a stupid question at all. It would be amusing to ask it on various forums (I came from math.stackexchange.com) and see the different types of response you get. –  Ross Millikan Feb 4 '13 at 5:14

6 Answers 6

up vote 15 down vote accepted

Pedigree collapse is the term used to describe how

reproduction between two individuals who knowingly or unknowingly share an ancestor causes the family tree of their offspring to be smaller than it would otherwise be

as it

collapses the binary tree into a directed acyclic graph with two different, directed paths starting from the ancestor who in the binary tree would occupy two places

(Both quotations from the linked wiki article on the International Society of Genetic genealogy website, which acknowledges material from the Wikipedia article on pedigree collapse.)

Pedigree collapse can arise for a number of reasons, the most obvious being:

  • cousins being encouraged to marry to keep wealth and property within a family
  • the frequent requirement in the past for royalty only to marry other royalty (to a lesser extent this may also apply to nobility)
  • in historical communities without access to modern transport (and modern isolated communities), marriages most frequently taking place between people living within walking distance
  • in countries with diverse immigrant communities of recent arrivals (such as 19th century America), the tendency of individuals to marry within own their own linguistic, ethnic or cultural group.

According to this article in Family Tree Magazine (March 2011) it has been estimated by one academic that 80% of historical marriages took place between second or closer cousins, and another has concluded that 86% of the residents of England in 1066 were ancestors to all residents of England in 1980.

So none of us will have to track down 1012 ancestors, even if the paper-trail and/or DNA evidence would allow us to so do.

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Thanks for your answer and for the link. I like the conclusion of the article ("Some geneticists believe that everybody on Earth is at least 50th cousin to everybody else.") a lot. –  Yann Feb 2 '13 at 18:59
    
@Yann: you might read The Seven Daughters of Eve on this. But as interesting as it is, it really considers only direct female lines. There is similar research on direct male lines, but many lines have both male and female links. –  Ross Millikan Feb 4 '13 at 5:28

Re: "If someone wanted to draw its family tree over the last 1000 years, he'd write a binary tree with a depth of about 40..."

No one would attempt to draw a tree of that depth. I appreciate that this is a theoretical exercise since there are no cases that I know of where anyone would claim a full genealogical history over that span. Some "royal" households may claim to be able to trace back a specific line that far but that's not a complete genealogy as many partners' lines will have been excluded.

As you say yourself, the depth of the tree is approximately Depth = Int(TimeSpan/Generation), or 1000/25 = 40 in your example. An equivalent binary tree would have 2(Depth+1)-1 people, or 241-1 (about 2,200,000,000,000) in your example, which is simply an unmanageable number. It would also be ignoring all siblings but that's because it would be a pedigree chart rather than a family tree.

There would certainly be a lot of pedigree collapse, and this might be far more common than one might expect. Our ancestors would have been part of communities - villages, hamlets, tribes, towns, etc - and this would increase the prospect of shared ancestors. Remember too that some religious communities strive to share ancestors and cultural heritage by deliberately marrying within their own community.

Although someone might draw their lineage as a tree (whether as a 'family tree' or a 'pedigree chart'), the internal relationships do not form a "tree" in mathematical terms. This is an important consideration for software designers since you don't want to be represented twice (with independent details), or end up being your own ancestor. Again, in mathematical terms, those relationships form something called a Directed Acyclic Graph, or DAG. This is more general than a simple (mathematical-)tree but still has rigid constraints, in contrast to a network where virtually any connections are deemed valid.

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Most of the public discussion on pedigree collapse doesn't, in my view, adequately deal with several issues:

  • In the past, people tended to marry people from nearby. The probability that these people were distant cousins was therefore higher than if they looked further afield to meet partners (this also applies to nobles marrying within their own class).
  • The number of children per family varied.
  • Not everyone had children.

To explore this, at least in small-sample form, I wrote a small program in Mathematica that simulates populations and their genealogies. The basic idea is that each person is located at a point between 0 and 1. The subset of surviving males and females are matched into couples, after shuffling one sex's locations within subgroups of the population. (I shuffle them in groups of n and then in groups of (1.2 n) so that lineages can migrate over the whole range of locations over a run of generations.) Each couple then has a random number of children. (For those who care, the number of children is capped at 20 and distributed according to a Zipf distribution with a mean of 2.6, or a bit over 1% population growth a year, compounded over 25 years. This growth rate can be varied in my program.)

From a starting population of 10,000, the distribution of unique ancestors depends heavily on how local couple matching is. As the histograms below show, if people match in groups of 100–120, there is much more pedigree collapse than if they match in groups of 500–600. Less than one-quarter of people have 32 unique great-great-grandparents in the first case, compared with around 60% in the second case.

enter image description here

Adding one more generation makes the pedigree collapse even more extensive.

enter image description here

This is a small-population case (any more takes a long time to compute) but it is clear that even in a larger total population with less-local matching of couples, some pedigree collapse for some lineages is inevitable even in the 5–10 generation span.

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I believe that pedigree collapse is not the correct - or at least not the only - answer to the problem or paradox of the ever growing number of ancestors.

  1. The number of ancestors does always grow, unless it is forced to decrease because of lack of partners.

  2. Even under modest assumptions about the collapse rate the number of ancestors will grow exponentially - only with a smaller base than 2.1

  3. So in any case the number of ancestors of an individual will eventually reach the number of all living humans at a past point in time - sooner or later.

Pedigree collapse does only delay this point in time. So you nevertheless face the problem of "as many ancestors as living humans".

Once you accepted that, it's quite easy to see a solution of the paradox: Beyond your "point of equality" (between ancestors and living humans) all humans have to be your ancestors.

So the question only is:

Under which assumptions can one's point of equality be calculated?

Another one:

Is there still a recursion law like (*) beyond the point of equality? How does it look like?


1 Consider the maybe too simplified recursion law for the number a(n) of ancestors in generation n+1:

a(n+1) = 2·a(n) - p·2·a(n) = 2·(1-p)·a(n) (*)

This means that each of one's ancestors in generation n has two parents diminuished by a certain percentage p of those parents that happen to be the same person. (This is in essence pedigree collapse.) It gives - for larger n - the number of ancestors

a(n) = (2·(1-p))n

which still is an exponential law.

With a constant collapse rate of p = 0.25 - which means that cousin marriage is the rule - the number of ancestors grows like 1.5n (compared to 2n) which still implies 10 millions ancestors after 40 generations.

Note, that for p = 0.5 (sibling marriage) there is no growth at all!

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1  
(1) Pedigree collapse doesn't describe how ancestral numbers stop growing, it describes how that growth is slower than would be expected in a simple binary tree without duplication of ancestors. (2) I am very confident that my family tree doesn't include all those humans who died without reproducing -- i.e. all humans alive at any calculated point are not my ancestors. –  ColeValleyGirl Jun 15 at 15:16
    
ad (1): All I wanted to say is that pedigree collapse doesn't explain a lot about ancestors outgrowing all living humans (by mere number). ad (2): I have to think about your argument: you are right that I do not descend of a human who died without reproducing. But probably of one of his ancestors? –  Hans Stricker Jun 15 at 17:50

I attempted an answer, based on a royal genealogical model (that of Margrethe II, current Queen of Denmark). I used 20 generations of data from the Roglo database, bringing her ancestry back to the 14th century. She has 2,507 known ancestors in that database out of a million lines. Of course, I had to estimate those lines that were missing from the database. Even European royals who are mostly of royal ancestry have outliers. For Margrethe and other Scandinavian royalty, that's some upper class French commoners related to the Bernadottes. In any case, her 20th generation ancestors are just a few thousand.

Then I extrapolated that line until it reached about 50% of the population (so as to not cover people with no descendants), and this correlates pretty well with Chang, Rohde, and Olson's more liberal model for the identical ancestors point. It makes sense, I think, that European royalty's ancestors would approach total human population fairly early, since they were in a well-connected part of the world (compared to, say, Tasmania) and had fairly diverse ancestry that connected to multiple parts of Europe. From then on back I just assumed that their ancestors remained about 50% of total population, then integrated under the curve, using the same method that Keyfitz did to find total human population throughout history.

I'm no professional at this, so I might have got something wrong, but various models I ran in Mathematica came out with between 6 and 8 billion ancestors for Margrethe since ~12000 BP (i.e., the whole Holocene). Of course, people with more varied ancestry (thinking here of North Americans with ancestry in multiple parts of Europe and Africa) are going to have their ancestors grow faster and those from isolated islands will have it grow much slower. But all eventually reach the IAP. So, I imagine, first guess, that most humans range between 5 - 10 billion ancestors in the Holocene... and of course, countless more that we share with extinct archaics, other apes, etc.

The video version is at How many ancestors do you have? if you'd like to see.

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Hi JM Ruby, welcome to genealogy.SE! Thank you for your answer and the video you made. –  lejonet yesterday
    
Unfortunately I have only one vote for your answer, your video deserves far more! –  lejonet yesterday
    
Thank you! The next one is on descent from Genghis Khan. –  JM Ruby yesterday
    
Thanks for the edit. This is my first SE answer, so I wasn't sure about the formatting. –  JM Ruby yesterday
    
It’s a great network, make sure to check out our tour. SE supports Markdown syntax. –  lejonet yesterday

Some normalization and greater/lower bounding can be based on world population estimates such as those at http://www.vaughns-1-pagers.com/history/world-population-growth.htm.

Everybody now (call it 7B, the 2011 estimate, because there's no need for a lot of decimal points) had to come from somebody then (say 400M in 1000 AD). Any logarithmic generation arithmetic (or predictive model) should "color inside the lines" with respect to total populations.

How far to fine-tune the arithmetic depends on what we want to predict.

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