CX

I was pondering the implications of identifying the number of ancestors of a given present individual going back x generations. It looks like a binary tree that would imply for x generations back in time there are 2^x ancestors. The problem is that results in absurdities like my having over a trillion ancestors from 40 generations ago (only about 1000 years or so), but there were far fewer than a trillion people on earth in that time frame. So what am I doing wrong?

Coragyps

The tree folds in on itself: fourth cousins marry all the time, without even knowing they share a couple of ancestors. (Back home in Arkansas, you might want to change "fourth" to "first.)
How the mathematics of that works, I have very little clue.

lpetrich

In an evenly-mixed population of N individuals, the number of generations to ancestor interbreeding will be log(2,N); one has to look back that number of generations for one's ancestors to have a significant probability of interbreeding.

Doubting Didymus

For a start, your assumption of a binary tree seems to assume that each member of each generation will have exactly two descendants. Thats the case only for things that reproduce by division, like bacteria and other single cellers. Humans have any number of children. Some have fifty, some have seven, some couples have four (the only option that would validate the binary tree model to a degree), some couples have 2 (note that if this were true for everyone the population would stay completely constant), and some people, like me, just hate kids and never want to go near them, let alone make some.

You've forgotten something else, too:

Coragyps

DD - you're conflating (is that the word?? maybe "confusing") descendants with ancestors: each of us really did have exactly two ancestors in the generation one step older, and four in the one before that. The flaw in the 2^x tree that CX asked about is that a single ancestor can be on more than one lineage.

CX

Ah! That makes sense. I'm chagrinned that didn't occur to me. I wonder then if there is an algorithmic way to determine the number of ancestors of a present individual going back x ancestors. I ask because I want to write a simulation.

CX

Another issue that seems relevant to me is the likelihood of one individual ancestor procreating with multiple partners.

Doubting Didymus

Damnation! I'm so used to dealing with half witted arguments about population increase indicating a 6000 year old earth that I trotted out the standard refutation without thinking. Let this be a lesson to the rest of you: reading too many creationist spiels rots your brain.

Undercurrent

There's probably no algorithmic way to do that without incorporating the population dynamics. I know for a fact that 30 generations ago, there had to have been a person from whom I am descended along two lines, since if not, I would have had more than a billion distinct ancestors among less than a billion people, but I expect that a lot or people alive at that time were not my ancestors (e.g. chinese people) and my tree starts folding over at 10 generations at most, just because of the way the gene pool splits.