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Pedigree collapse: why no one has a billion ancestors

Why the doubling of ancestors cannot go on: pedigree collapse worked through with small numbers, the Spanish Habsburgs, and what Chang and Rohde found.

Portrait of Ruth AshdownRuth Ashdown · · 10 min read
Pedigree collapse: why no one has a billion ancestors

You have 2 parents, 4 grandparents and 8 great-grandparents. Keep doubling and the numbers soon stop making sense. This piece follows the doubling until it breaks, then looks at what happens instead: the same people turn up in more than one place in the tree. Genealogists call this pedigree collapse.

The doubling, and where it breaks

Number the generations upward from you: your parents are generation 1, your grandparents generation 2, and so on. Generation g has 2^g places to fill. To turn generations into years we need a generation length; the table uses 30 years, a round figure that Rohde, Olson and Chang also use in one of their illustrations. Change it to 25 or 35 and the conclusion stays the same, only sooner or later.

Generation back Places in the tree (2^g) Roughly how many years back
1 2 30
5 32 150
10 1,024 300
15 32,768 450
20 1,048,576 600
25 33,554,432 750
30 1,073,741,824 900
37 137,438,953,472 1,110

Now set those numbers beside some estimates from the Population Reference Bureau. Its demographers put the world's population at about 300 million in AD 1 and about 450 million in 1200. At generation 30, about 900 years back for someone born today, a family tree has more than a billion places, over twice the PRB figure for the entire world in 1200. By generation 37 the tree has more places than the roughly 117 billion people PRB estimates have ever been born.

Since the people did not exist, the places must be shared. Somewhere above you, one person fills two places, or two hundred, or two million.

How a tree folds: one cousin marriage

The term pedigree collapse covers any case where a person appears more than once in a family tree, so that the number of distinct ancestors falls below 2^g. Older synonyms are implex and the German Ahnenschwund, "loss of ancestors."

The smallest useful example is a child whose parents are first cousins. Call the child X and the parents P and Q. P and Q share a pair of grandparents; call them the shared couple. For X, that couple are great-grandparents, and they appear twice: once through P and once through Q.

Generation above X Places Distinct people People who appear twice Shortfall
1, parents 2 2 0 0%
2, grandparents 4 4 0 0%
3, great-grandparents 8 6 2 25%
4, great-great-grandparents 16 12 4 25%
5, the generation above that 32 24 8 25%

Work the rows by hand. In generation 3 the shared couple fill 4 of the 8 places, so 8 places hold 6 people. In generation 4, the shared couple's own 4 parents each appear twice, so 16 places hold 12 people. Every generation above that, the duplicated block doubles along with everything else, so the shortfall stays at one quarter. The table assumes there are no other loops; real trees usually have several, and each one takes its own bite.

Our pedigree collapse calculator shows the number of places at any generation, sets it beside the world's population at the time, and turns the number of distinct people you have found into a share of repeats.

Counting ancestors is not measuring inbreeding

It is tempting to treat the shortfall as a measure of how inbred someone is. The biologist Raymond Pearl proposed something close to that in a series of papers between 1913 and 1917: compare the actual number of distinct ancestors in each generation with the maximum possible.

Sewall Wright's 1922 paper on inbreeding and relationship opens by pointing out the weakness of that approach, one Pearl himself had acknowledged: very different breeding systems can give the same counts. Wright's example is neat. In a line where double first cousins keep marrying each other, an animal has 2 parents, 4 grandparents and then only 4 ancestors in every earlier generation. An animal bred by crossing two separate lines, each kept going by brother and sister matings, has exactly the same counts. Yet the first will, given enough generations, carry two identical copies of nearly every gene, while the second carries two different copies at as many genes as possible. Same shortfall, opposite outcome.

The reason is distance. What matters is how short the loops are, and how close to the person they close, and that is what Wright's inbreeding coefficient F measures by adding up paths. We work through the method in the coefficient of relationship, worked out by hand. For the child of first cousins, F = 1/16. Wright also made the point that matters most to family historians: "Remote common ancestors in general have little effect on the coefficient."

A tree that folded hard: the Spanish Habsburgs

Royal houses that married mostly among themselves are the textbook cases. Gonzalo Alvarez, Francisco Ceballos and Celsa Quinteiro (2009) calculated inbreeding coefficients for the Spanish Habsburg kings from a pedigree 16 generations deep with more than 3,000 individuals.

The last of those kings, Charles II, was born in 1661 to Philip IV and Mariana of Austria, who had married in 1649. Mariana was Philip's niece: her mother was his sister. So Philip's parents, Philip III and Margaret of Austria, appear in Charles's tree twice, as his father's parents and as his mother's mother's parents. Count the places in generations 2 and 3 together: 4 + 8 = 12. Two people fill four of them, so at most 10 different people fill those 12 places, and that is before any of the older loops.

The path method gives the inbreeding expected from that one marriage. Uncle and niece share two ancestors, his parents. The uncle is 1 generation below them and the niece 2, so each path is worth (1/2)^(1 + 2 + 1) = 1/16, and two paths make F = 1/8 = 0.125.

The study's figure for Charles II was 0.254, practically twice that, and close to the 0.25 expected for the child of a brother and sister or of a parent and child. The extra came from the many older loops further up the tree. Across the dynasty, the coefficient rose from 0.025 for Philip I, the founder, to Charles's 0.254. The authors also found that inbreeding lowered the chance that the kings' children survived to age 10, and they conclude that inbreeding played a large part in the end of the dynasty, which came when Charles died without children in 1700.

Every tree folds, and sooner than you might think

Royal trees fold fast, but every tree folds. Joseph Chang (1999) worked out how fast in an idealized model: a population of n people who choose partners at random, generation after generation. Two results stand out.

First, the most recent common ancestor of everyone alive, the latest person who is an ancestor of all of them, lived about log2(n) generations ago. Second, at about 1.77 × log2(n) generations back, every person then alive is either an ancestor of everyone living now or an ancestor of no one living now. Rohde, Olson and Chang call this the identical ancestors point.

Try it with n = 1,000,000. Since 2^20 = 1,048,576, log2 of a million is just under 20. So the common ancestor lived about 20 generations back, and the identical ancestors point comes at about 1.77 × 20 ≈ 35 generations. The doubling table explains the first number: 20 generations is roughly where a single tree has as many places as the population has people. Rohde and colleagues use the same example, a randomly mating population of one million, and with 30-year generations they put that common ancestor around AD 1400.

Random mating is a big simplification, and Rohde, Olson and Chang (2004) set out to test how much it matters. Their more realistic simulation included geography and slow migration. In one conservative version it put the most recent common ancestor of all living humans at a mean date of 1,415 BC and the identical ancestors point at 5,353 BC. Allowing more migration moved those to AD 55 and 2,158 BC. The authors are careful to say the date cannot be fixed with precision. What holds up across their models is that it falls within the last few thousand years.

Genealogical ancestors and genetic ancestors

None of this means you carry DNA from all those people. DNA passes down in long segments, and segments drop out along the way, so far enough back most of your genealogical ancestors left you no DNA at all. Rohde and colleagues note that a person inherits little or nothing from the vast majority of the ancestors living at the identical ancestors point.

Peter Ralph and Graham Coop (2013) measured the genetic side directly, using long shared stretches of DNA in 2,257 Europeans. A pair of people from neighboring populations shared around 2 to 12 genetic common ancestors from the last 1,500 years, and upwards of 100 from the 1,000 years before that. Those genetic ancestors, they point out, are a tiny fraction of the genealogical ones: even people from opposite ends of Europe are expected to share millions of common genealogical ancestors over the last 1,000 years.

Questions family historians ask

Does pedigree collapse mean my family was inbred?

Not by itself. Every tree collapses if you go back far enough; that is arithmetic. What matters for F is a loop that closes within a few generations, such as a marriage between first or second cousins. Many communities have long preferred marriages between particular cousins, as our explainer on cross and parallel cousins describes, and those loops show up quickly in a tree.

How do I measure it in my own tree?

For each generation, count the distinct people you have identified and compare with 2^g. Keep in mind that a duplicate at one level repeats in every generation above it, since the same person brings the same parents. If you enter each person once and link them into the tree wherever they belong, rather than typing them in twice, the duplicates are easy to count, and the calculator will turn your count into a percentage.

Am I descended from someone famous who lived a thousand years ago?

In Chang's model, beyond the identical ancestors point, anyone with living descendants is an ancestor of everyone. Real populations are lumpier, as Rohde and colleagues stress, so the answer depends on where your ancestors lived and how far back you go. But if the person lived many centuries ago in a region your family comes from, and has descendants alive today, you have a real chance of being one of them, and the further back they lived, the better the chance. Proving the line with documents is a different matter.

What does a smaller loop look like?

Take the child of second cousins. The parents share a couple who are the child's great-great-grandparents, generation 4. That couple fills four of the 16 places, so there are 14 distinct people, a shortfall of 2/16 = 12.5 percent, and the shortfall stays at 12.5 percent in every generation above. The inbreeding coefficient is much smaller than for first cousins: two paths, each with 3 generations on each side, give F = 2 × (1/2)^(3 + 3 + 1) = 2/128 = 1/64.

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