Pedigree collapse
A Socratic walk-through of pedigree collapse — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why does doubling your ancestors each generation back soon demand more people than ever lived?
Everyone has two parents. Those parents each had two, so four grandparents; eight great-grandparents; sixteen beyond that. The rule feels unimpeachable — it is not a theory about history, just the definition of having parents — and it can be run backwards as far as you like.
Run it far enough and it produces a number larger than the human race. Somewhere a few centuries back, your family tree demands more ancestors than there were people alive to be them. Nothing in the reasoning was wrong, and nobody is missing a parent. So what exactly broke?
Reasoning it through
REASONING #First, get the growth honest, because the surprise depends entirely on how fast it moves. Ten generations back the tree asks for 1,024 slots — unremarkable, roughly a large wedding. Twenty generations asks for about a million. Thirty asks for about a billion. The number is not creeping; it is multiplying by a thousand every ten generations, and our intuitions handle multiplication badly over long runs.
Now hold that against the other quantity. At about thirty years to a generation, thirty generations back lands somewhere near the twelfth century. Estimates of world population then are in the region of three hundred million — and that is everyone on the planet, most of them on continents your family had no contact with, and only a fraction of them of an age to be anyone's parent.
So the doubling curve, which begins far below the population line, crosses it. Ask when: a billion slots against a few hundred million people means the crossing happens a little before thirty generations, in the neighbourhood of eight centuries ago. That is not deep prehistory. That is the era of cathedrals.
Something must give, so ask carefully which of the two claims is false. Is it "everyone has two parents"? No — that survives. Is it the arithmetic? No. Then the false step is the one that was never stated: we quietly assumed the slots were filled by distinct people.
They are not. The tree is a count of positions, not of persons, and one person may occupy many positions at once. Every time two people who share an ancestor have a child — second cousins, fourth cousins, cousins so distant that neither family has ever heard of the link — two branches of the tree that were drawn as separate lines fold into the same individual. Two slots become one person. Do that repeatedly and the tree stops widening.
Would this be rare? Consider where people actually found partners for most of history: within a village, a valley, a parish, a caste, a linguistic community — pools of hundreds or a few thousand, not billions. In a pool that size, a doubling tree exhausts the distinct candidates within a handful of generations, and after that every additional generation must reuse people. Collapse is not an occasional oddity introduced by unusual marriages; it is the default consequence of a tree growing faster than the population it draws from.
And once you see the direction, the consequence follows. If distant lines keep folding back together, then a person alive long enough ago can sit in an enormous number of positions in one tree — and, by the same token, in the trees of enormous numbers of living people. Chang's modelling work in 1999, and a follow-up by Rohde, Olson and Chang in 2004 that added realistic migration between regions, put the most recent common ancestor of everyone alive today only a few thousand years back, with a still earlier point at which every person then living is either an ancestor of all of us or of none of us. Those are model results, sensitive to their migration assumptions rather than measured from records, so treat the specific millennia as estimates rather than facts.
The analogy
THE ANALOGY #Think of a hall of mirrors where you count reflections. Two mirrors give you two images, then four, then eight, and the sequence is perfectly correct as a count of images. It would be a mistake to conclude that the hall contains a thousand different people. The doubling is real; the people are the same few, seen from more and more angles.
Mirror images are copies of one original standing in the room now, whereas your repeated ancestors were genuinely separate lives that happen to be reached by more than one route through the tree — the duplication is in the paths, not in the person.
Clarifying the model
THE MODEL #Three things are easy to get slightly wrong here.
The first is to treat collapse as a claim about inbreeding in the loaded sense. It is not. Two people descended from a shared ancestor eight generations back share a vanishing fraction of genome by that route, and nothing about the health implications of close-kin marriage carries over. Collapse describes the shape of a pedigree, not the genetics of a couple.
The second is to assume that being an ancestor of everyone means contributing genes to everyone. It does not. You inherit roughly half your DNA from each parent, and at each step the ancestral contributions are chopped and sampled, so past about ten generations most of your genealogical ancestors have passed you no detectable DNA at all. Genealogical ancestry and genetic ancestry come apart, and the collapse argument speaks only to the first.
The third is to over-read the recency of common ancestry as a claim that populations were well mixed. They were not, and that is exactly why the models needed migration built in. All it takes is a thin trickle — an occasional traveller, a port, a border marriage — to connect otherwise separate pools, and the doubling then propagates the link through everyone downstream with startling speed. The result is not that human groups were similar, but that the ancestor graph is far more connected than the family-tree picture suggests.
A picture of it
THE PICTURE #How to readBoth quantities are shown as powers of ten, so each step up the axis is a tenfold increase. The bars are the ancestor slots the doubling rule demands; the line is a rough estimate of the entire world population at that date. The bars pass the line somewhere before thirty generations, and everything to the right of that crossing has to be filled by reusing people.
What became clearer
WHAT CLEARED #The doubling rule is not wrong; it is a rule about positions in a diagram, and we silently read it as a rule about persons. Every marriage between even remotely related people folds two positions onto one life, and in the small mating pools of actual history that folding is constant. A family tree is therefore not a widening triangle but a net that spreads for a while and then draws back in on itself — which is also why the ancestors of any two strangers overlap far sooner than either would guess.
Where to go next
ONWARD #- How genealogical ancestry and genetic ancestry diverge with depth, and why a documented ancestor may have left you no DNA.
- What the coefficient of relationship actually measures, and why distant cousin marriage barely moves it.
- How isolated populations — islands, mountain valleys — show collapse in an exaggerated form, and what that does to their genetics.
Key terms
TERMS #| Term | What it means |
|---|---|
| Pedigree collapse | the folding of a family tree when one individual fills more than one ancestral position, because the parents shared an ancestor. |
| Most recent common ancestor | the latest individual from whom every member of a given group descends. |
| Identical ancestors point | the date before which every person then living is an ancestor of everyone alive today, or of no one. |
| Mating pool | the set of people among whom partners were realistically found, usually far smaller than the world population. |
Every term the collection defines is gathered in the glossary.