Marius Comper

Genealogical combinatorics & historical demography

The Tree That Closes

Every person has 2 parents, 4 grandparents, and 8 great-grandparents. Doubling backward in time, by generation 30 a family tree demands 1,073,741,824 simultaneous ancestors — more than triple the entire population of Earth in the year 1100. Mathematically, a family tree is not an ever-expanding fan, but a woven spindle: past generation 20, lineages reconnect so frequently that every person on a continent shares the exact same ancestors.

1,073,741,824 theoretical slots in a single tree at generation 30 (~year 1120)
320 million estimated total population of Earth in the year 1100
Generation 33 where tree slots exceed today’s global population (8.15 billion)
1,000 years the horizon beyond which anyone with living descendants is an ancestor to everyone

The Duplication Paradox

The elementary logic of ancestry seems straightforward: with every step back, the number of ancestors doubles. Ten generations ago, around the year 1720, a person has 1,024 tree slots. Twenty generations ago, around 1420, there are 1,048,576 slots. Thirty generations ago, during the Crusades (around 1120 AD), the formula 230 produces 1,073,741,824 people.

Here arithmetic collides with physical reality. In 1100 AD, all of Europe had roughly 62 million people, and the entire globe held only 320 million. By generation 40 (circa 820 AD, the era of Charlemagne), the formula demands 1.1 trillion simultaneous ancestors — four thousand times the human population of the world at the time.

The mathematical consequence is inescapable: ancestor slots are not unique individuals. The same people reappear on thousands, then millions of different branches. This phenomenon is known in population genetics as pedigree collapse (or implex). A family tree does not expand forever; it chokes and tapers like a spindle, constrained by the envelope of real historical population.

The Ancestry Spindle

Explore how your ancestral tree bends and collapses across 40 generations

Jump to:
Generation / Estimated Year Generation 30
Gen 1 (1996) ~1126 Gen 40 (826)
Mating Pool & Geographic Mobility

Tighter endogamy accelerates pedigree collapse and narrows the pool of distinct ancestors earlier.

Theoretical slots (2g)
Estimated unique ancestors
Historical European population
Generation & Epoch
Generation 30 · Year ~1126
Theoretical tree slots
1.07 billion
Exactly 1,073,741,824 theoretical slots (2^30)
Estimated unique ancestors
~45 million
Distinct real ancestors in European model
Pedigree collapse ratio (Implex)
95.80%
Over 95% of tree slots are duplicate entries of already present ancestors.
Average repetition multiplier
23.8×
How many times an average ancestor appears across distinct branches of your tree.
Comparison to era population
Exceeds Europe's population in that year by 17.3× (62 million people)

Why We Are All Cousins

In 1999, Yale mathematician Joseph Chang proved that in any standard mixing population, the Most Recent Common Ancestor (MRCA) of all present-day individuals lived only dozens of generations ago. For Europe, genomic and mathematical models place this shared ancestor roughly 25 to 30 generations back — around 1000–1100 AD.

Even more startling is the Identical Ancestors Point (IAP), reached around 35 to 40 generations ago (the 9th–10th centuries AD). At that point in history, every single person who lived and left living descendants today is an ancestor of every single modern European. If a 10th-century peasant in Transylvania, a blacksmith in Germany, or a noble in France has even one descendant alive in the 21st century, they are an ancestor to everyone on the continent.

Genomic sequencing led by Peter Ralph and Graham Coop in 2013 (analyzing 2,257 genomes across 40 European populations) physically confirmed what combinatorics demanded: any two individuals chosen at random across thousands of miles share thousands of recent common ancestors and identical inherited chromosome blocks from the last millennium.

Generational Progression & Collapse Milestones

Comparative reference table for key generations, contrasting theoretical slots against historical population estimates.

Generation Approx. Year Theoretical Slots (2g) Europe Population World Population Collapse Status
Gen 1 1996 2 725M 5.84B 0% (All unique)
Gen 6 1846 64 250M 1.20B < 0.01%
Gen 10 1726 1,024 135M 680M ~0.1%
Gen 15 1576 32,768 95M 500M ~2.5%
Gen 20 1426 1,048,576 65M 380M ~40%
Gen 21 1396 2,097,152 60M 375M Exceeds regional population (~1.2M)
Gen 27 1216 134,217,728 70M 365M Exceeds European population (~70M) by 1.9×
Gen 30 1126 1,073,741,824 62M 320M Exceeds global population (~320M) by 3.3×
Gen 33 1036 8,589,934,592 58M 305M Exceeds today’s global population (8.15B)
Gen 40 826 1,099,511,627,776 45M 240M Identical Ancestors Point (>99.99% collapse)

Methodology & Sources

The number of theoretical slots in a family tree at generation g is calculated purely combinatorially via the binary power S(g) = 2g, setting g = 0 for the present individual.

The number of distinct ancestors U(g) is modeled using the standard recurrence for branching processes in finite populations (Chang 1999): U(g) = N · (1 - e-2·U(g-1)/N), where N represents the effective size of the ancestral mating pool, scaled by historical reproductive survival rates of ~79.7% per the Galton-Watson distribution.

Historical European and global population estimates are sourced from the Maddison Project Database (2020) and Colin McEvedy & Richard Jones (Atlas of World Population History, Penguin, 1978).