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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
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2 Answers

up vote 12 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
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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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