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How many ancestors do I have? Counts for 10, 12 and 20 generations

You have 2^n ancestors n generations back and 2^(n+1) - 2 in all: 1,024 at 10 generations, 4,096 at 12, over a million at 20. Why fewer real people fill them.

Portrait of Elspeth GilchristElspeth Gilchrist · · 9 min read
How many ancestors do I have? Counts for 10, 12 and 20 generations

Count back n generations and you have 2^n ancestors in that generation, and 2^(n+1) - 2 ancestors in all the generations from your parents up to it. That gives 1,024 ancestors 10 generations back (2,046 counting every generation in between), 4,096 at 12 generations (8,190 in all) and 1,048,576 at 20 generations (2,097,150 in all). Those figures count places in a family tree, though, and the number of different people who fill the places is smaller, often much smaller.

Where the two formulas come from

Everyone has two parents, so each step back doubles the count: 2 parents, 4 grandparents, 8 great-grandparents. Call your parents generation 1, and the count at generation n is 2 multiplied by itself n times, written 2^n.

The running total needs one more line:

2 + 4 + 8 + ... + 2^n = 2^(n+1) - 2

Here is a way to see it without algebra. Every generation holds exactly 2 more people than all the generations below it put together: your 8 great-grandparents outnumber your 2 parents and 4 grandparents, 6 people, by 2. So the total from generation 1 up to generation n is always the next generation's count, 2^(n+1), minus 2. Check it at n = 3: 2 + 4 + 8 = 14, and 2^4 - 2 = 16 - 2 = 14.

Ancestors by generation, 1 to 20

The last column turns generations into years with a generation length of 30 years. That figure comes from Marc Tremblay and Hélène Vézina (2000), who measured the gaps between parents and children in the BALSAC population register of Quebec and concluded that a mean of 30 years fits better than the 20 or 25 years often used in population genetics. They also found marked differences between lines through fathers and lines through mothers, so treat the column as a rough guide.

Generation Who they are Ancestors in this generation All ancestors up to here About how long before your birth
1 parents 2 2 30 years
2 grandparents 4 6 60 years
3 great-grandparents 8 14 90 years
4 great-great-grandparents 16 30 120 years
5 3rd great-grandparents 32 62 150 years
6 4th great-grandparents 64 126 180 years
7 5th great-grandparents 128 254 210 years
8 6th great-grandparents 256 510 240 years
9 7th great-grandparents 512 1,022 270 years
10 8th great-grandparents 1,024 2,046 300 years
11 9th great-grandparents 2,048 4,094 330 years
12 10th great-grandparents 4,096 8,190 360 years
13 11th great-grandparents 8,192 16,382 390 years
14 12th great-grandparents 16,384 32,766 420 years
15 13th great-grandparents 32,768 65,534 450 years
16 14th great-grandparents 65,536 131,070 480 years
17 15th great-grandparents 131,072 262,142 510 years
18 16th great-grandparents 262,144 524,286 540 years
19 17th great-grandparents 524,288 1,048,574 570 years
20 18th great-grandparents 1,048,576 2,097,150 600 years

The names follow the usual genealogical pattern. After great-great-grandparents, the number in front of "great" is the generation minus 2, so generation 10 holds your 8th great-grandparents. Some genealogists already write "2nd great-grandparents" for generation 4; it is the same set of 16 people.

The three numbers people ask about

How many ancestors in 10 generations?

Your tenth generation, the 8th great-grandparents, has 1,024 places. Add every generation from your parents up and you get 2,046. At 30 years a generation they were born about 300 years before you, so for someone born in 1990, around 1690. At 25 years a generation the date moves forward to about 1740.

How many ancestors in 12 generations?

The twelfth generation, your 10th great-grandparents, has 4,096 places, and the total from your parents up is 8,190. Half of them, 2,048, sit on your father's side and half on your mother's. Twelve generations of 30 years is 360 years: around 1630 for someone born in 1990.

How many ancestors in 20 generations?

The twentieth generation alone has 1,048,576 places, just over a million, filled by your 18th great-grandparents. The total from your parents up is 2,097,150. At 30 years a generation that is about 600 years back, the late 1300s for someone born in 1990.

A million is still smaller than the world's population at the time, so the doubling has not yet reached the impossible. It gets there a few generations later. Our pedigree collapse calculator sets the doubling beside Our World in Data's population estimates, and at its default settings (born 1990, 30-year generations) the places outnumber everyone alive by generation 29, around 1120.

Two conventions that change the answer

Before comparing your figure with someone else's, check two things.

First, where counting starts. This piece calls your parents generation 1. Some people count themselves as generation 1 instead, and then "10 generations" ends at your 7th great-grandparents: 512 people in the top row and 1,022 ancestors in all, or 1,023 names if you include yourself. Neither convention is wrong, but the two answers differ by a factor of two.

Second, whether a question means one side of the family. Ancestors on your father's side alone number 2^(n-1) in generation n, half the full count, and the same for your mother's side.

A numbering system that does the arithmetic for you

Genealogists have long numbered ancestors with a scheme called the ahnentafel (German for "ancestor table"), also known as Sosa or Sosa-Stradonitz numbering. Michaël Eytzinger set out its principle in 1590, Jerónimo de Sosa popularized it in 1676, and Stephan Kekulé von Stradonitz refined it in 1898. The rule fits in one line: you are number 1, and the father of person k is 2k and the mother is 2k + 1.

So your father is 2 and your mother 3; your father's parents are 4 and 5; your mother's parents are 6 and 7. Generation n uses the numbers from 2^n up to 2^(n+1) - 1, which gives two checks on the table above. There are 2^n numbers in that range, the count for the generation. And the largest number used by generation n is 2^(n+1) - 1, which is every ancestor up to that generation plus you.

The numbers also encode the route. Write an ancestor's number in binary, drop the leading 1, and read each remaining 0 as "father" and each 1 as "mother". Take ancestor 300. Since 2^8 = 256 ≤ 300 < 512, this person is in generation 8, one of your 6th great-grandparents. In binary, 300 is 100101100. Dropping the leading 1 leaves 00101100, which reads in kin-type notation (F father, M mother, read outward from you) as FFMFMMFF: your father's father's mother's father's mother's mother's father's father. Check it by halving and dropping any remainder: 300, 150, 75, 37, 18, 9, 4, 2, 1. Read from the 1 upward, every even number is a father and every odd one a mother.

Why the real number of different people is smaller

Every place in the tree needs a person, but nothing stops one person from filling two places. When that happens genealogists speak of pedigree collapse.

The ahnentafel shows it cleanly. Wikipedia's article on the system prints the ancestor table of Prince William, and Queen Victoria appears in it twice. She is number 79 through his grandfather Prince Philip, whose mother, Princess Alice of Battenberg, was a granddaughter of Victoria's daughter Princess Alice. She is also number 81 through his grandmother Elizabeth II, whose father, grandfather and great-grandfather were George VI, George V and Edward VII, Victoria's son. Run the rule: Elizabeth is 5, George VI is 10, George V is 20, Edward VII is 40 and his mother is 2 × 40 + 1 = 81. Both 79 and 81 lie between 64 and 127, generation 6. Victoria's husband, Prince Albert, takes the numbers just below hers, 78 and 80.

So William's sixth generation has 64 places and, from this loop alone, at most 62 different people. Enter generation 6 and 62 people in the calculator and it reports 3.125% collapse, which is 2 repeats out of 64. The couple bring their own parents with them, so the same share carries upward to every earlier generation unless other loops add to it.

Royal houses are the famous cases, but ordinary trees fold for an ordinary reason: people married within reach. Imagine that half your lines lead back to one valley whose parishes held 3,000 adults in the generation that is 12 back for you. That half of your tree has 2,048 places at generation 12, 4,096 at generation 13 and 8,192 at generation 14, while the valley, in this example, stays about the same size. Long before the places outnumber the people, cousins of various degrees will have married, and the same couples start appearing in several places. Our explainer on pedigree collapse follows a single cousin marriage through the generations above it.

How many of them passed you any DNA?

That is a separate question with a smaller answer again. DNA reaches you in chunks, not as an even blend of everyone. The geneticist Graham Coop worked out a rough count in 2013. Going back k generations, the autosomes (the 22 non-sex chromosomes that cousin matching compares) you received from one parent are broken into about 22 + 33 × (k - 1) chunks, using 22 autosomes and about 33 crossovers per generation. Doubling that for both parents sets a rough ceiling on how many ancestors in generation k can have contributed, since each contributing ancestor needs at least one chunk. Coop also notes that by about 9 generations back it becomes quite likely that a given ancestor left you no autosomal DNA at all.

Here is his formula set against the counts above:

Generation Ancestor places (2^n) Chunks of your DNA from that generation, about 2 × (22 + 33 × (n - 1))
10 1,024 638
12 4,096 770
20 1,048,576 1,298

At 20 generations, at most about one place in 800 can hold someone whose DNA you carry, since 1,048,576 / 1,298 is about 808. The rest are still your ancestors in every genealogical sense; they just left no trace a DNA test can find.

To see these totals for your own birth year and a generation length of your choosing, the pedigree collapse calculator does the doubling, and if you know how many different people you have found in a generation, it turns them into a share of repeats.

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