Unique Ancestors

A unique ancestor is simply a person who is an ancestor, notwithstanding that they may have contributed to our descent in more than one direct line.

I present herewith a hypothesis for a distribution curve that may give us a clue in the quantisation of the unique ancestral counts. There is a complex relationship between the total ancestors we have in a binary tree sense and those who are unique.

For some time now I have been interested in the mathematics of ancestry. My interest was triggered by the early April 1997 discussion thread in soc.genealogy.medieval on the counts of ancestors to a particular generation, and the expectation of the number of unique ones that count included. Subsequently, I have found that other people share this same interest, and who have followed similar lines of analysis to mine.

There appeared to be a strange divergence in examples previously given between the growth rates of total and unique ancestors that warranted further examination. I therefore wrote a small computer program of my own to assist in the study of these effects. For a specific individual in a database, the program counts the recorded ancestral links by the unique people involved. I have used this information to extrapolate an empirical curve for unique ancestors. Another programmer has also written a program to perform these same calculations, reference to which is shown below.

I emphasise that my "research" is basically empirical, not theoretical. I am also not an expert in mathematics, merely an interested observer and applicator. The phenomenon I describe must have previously been noted, and perhaps there are other explanations of which I am unaware.

For the purpose of these notes, I have defined a few special terms used in the table, as follows.

The following details for Prince William of Wales from my own database cover the range to 80 generations (including the data of some legendary or mythical people), but it serves to illustrate the effects involved. Appropriate figures are shown in scientific notation for brevity. Whether or not we know exactly who our ancestors were by name, we can still be sure that people existed in those ancestral roles!

To Generation	Total Ancestors	Recorded Knowns	Recorded Uniques	Estimated Uniques	Highest # Descents
1	0	0	0	0	0
2	2	2	2	2	1
3	6	6	6	6	1
4	14	14	14	14	1
5	30	30	30	30	1
6	62	62	62	62	1
7	126	126	122	122	2
8	254	253	236	237	2
9	510	482	422	445	4
10	1,022	885	723	827	6
11	2,046	1,589	1,191	1,508	12
12	4,094	2,792	1,861	2,649	17
13	8,190	4,978	2,858	4,517	28
14	16,382	8,973	4,301	7,476	45
15	32,766	16,369	6,356	11,962	77
16	65,534	29,943	8,917	18,216	193
17	131,070	55,278	12,191	26,685	354
18	262,142	102,698	16,062	37,385	694
19	524,286	191,703	20,305	49,882	1,286
20	1.0E+06	359,851	24,649	63,427	2,600
21	2.1E+06	676,348	28.708	76,874	5,906
22	4.2E+06	1.3E+06	32,668	90,904	8,384
23	8.4E+06	2.4E+06	36,414	105,207	14,106
24	1.7E+07	4.4E+06	39,759	119,050	26,759
25	3.4E+07	8.1E+06	42,568	131,761	42,097
26	6.7E+07	1.5E+07	45,001	143,975	80,021
27	1.3E+08	2.7E+07	47,038	155,492	155,445
28	2.7E+08	4.8E+07	48,706	166,222	245,424
29	5.4E+08	8.4E+07	49,993	175,696	370,665
30	1.1E+09	1.5E+08	50,915	183,510	874,387
31	2.1E+09	2.6E+08	51,615	190,425	1.6E+06
32	4.3E+09	4.4E+08	52,139	196,590	2.2E+06
33	8.6E+09	7.4E+08	52,497	201,734	3.3E+06
34	1.7E+10	1.2E+09	52,757	206,397	5.2E+06
35	3.4E+10	2.0E+09	52,941	210,607	8.8E+06
36	6.9E+10	3.1E_09	53,076	214,645	1.6E+07
37	1.4E+11	4.8E+09	53,188	219,141	2.6E_07
38	2.7E+11	7.3E+09	53,282	224,329	4.3E+07
39	5.5E+11	1.1E+10	53,370	231,146	7.4E+07
40	1.1E+12	1.6E+10	53,451	240,135	1.1E+08
41	2.2E+12	2.3E+10	53,522	251,694	1.8E+08
42	4.4E+12	3.2E+10	53,595	269,637	3.5E+08
43	8.8E+12	4.3E+10	53,660	294,491	5.7E+08
44	1.8E+13	5.7E+10	53,715	328,081	8.0E+08
45	3.5E+13	7.5E+10	53,754	366,894	1.0E+09
46	7.0E+13	9.7E+10	53,788	422,685	1.7E+09
47	1.4E+14	1.2E+11	53,823	517,611	2.3E+09
48	2.8E+14	1.5E+11	53,852	647,288	2.8E+09
49	5.6E+14	1.9E+11	53,878	839,235	3.2E+09
50	1.1E+15	2.4E+11	53,900	1,110,154	4.0E+09
51	2.3E+15	2.9E+11	53,916	see notes below	5.1E+09
52	4.5E+15	3.5E+11	53,927		6.3E+09
53	9.0E+15	4.3E+11	53,937		7.2E+09
54	1.8E+16	5.0E+11	53,946		7.7E+09
55	3.6E+16	5.9E+11	53,951		8.5E+09
56	7.2E+16	6.7E+11	53,959		9.2E+09
57	1.4E+17	7.5E+11	53,967		1.2E+10
58	2.9E+17	8.2E+11	53,972		1.3E+10
59	5.8E+17	8.8E+11	53,978		1.6E+10
60	1.2E+18	9.2E+11	53,985		1.8E+10
61	2.3E+18	9.6E+11	53,991		1.9E+10
62	4.6E+18	1.0E+12	53,999		1.9E+10
63	9.2E+18	1.0E+12	54,007		2.0E+10
64	1.8E+19	1.1E+12	54,015		2.0E+10
65	3.7E+19	1.1E+12	54,024		2.1E+10
66	7.4E+19	1.1E+12	54,034		2.1E+10
67	1.5E+20	1.1E+12	54,043		2.1E+10
68	3.0E+20	1.1E+12	54,051		2.2E+10
69	5.9E+20	1.2E+12	54,056		2.2E+10
70	1.2E+21	1.2E+12	54,059		2.2E+10
71	2.4E+21	1.2E+12	54,063		2.2E+10
72	4.7E+21	1.2E+12	54,068		2.2E+10
73	9.4E+21	1.2E+12	54,073		2.2E+10
74	1.9E+22	1.3E+12	54,077		2.2E+10
75	3.8E+22	1.3E+12	54,081		2.2E+10
76	7.6E+22	1.3E+12	54,085		2.2E+10
77	1.5E+23	1.4E+12	54,089		2.3E+10
78	3.0E+23	1.4E+12	54,091		2.3E+10
79	6.0E+23	1.4E+12	54,094		2.3E+10
80	1.2E+24	1.5E+12	54,096		2.3E+10

This high level of detail is shown to emphasise the magnitude of the numbers involved. At 80 generations (about 2000 years) we all have descended from 1.2E+24 total ancestors. This is a VERY BIG number. We all know for instance that the distance travelled by light in a year is a large number, but if put into perspective, the number of millimetres (a small but practical unit) travelled in a light year is only 9.5E+15. Compare this to 1.2E+24. Wow!

Since 1997 I have been researching and recording the ancestors of Prince William with kind assistance from a number of other people. In this time his ancestors in my database have grown from 13,000 to about 54,100 currently. In 1998 at the 40 generation point my records showed 21,400 out of a database total of 22,200. At that time my 40 generation projection of Estimated Uniques was 185,220. Compare that to now, where with 53,451ancestors at that same point the projection is 240,135. Considering the huge change in the number of people recorded, there is still a quite similar result for the projection.

The data we have is in effect a statistical sample, albeit a biased one. The greater the sample size, the higher the reliability of the information. In the above case, the percentages of Recorded Knowns to Total Ancestors at each 5 generation step are 100%, 78%, 44%, 32%, 22%, 12%, 4%, 1% and really tiny after 40 generations. The most recent generations therefore provide more reliable and complete data. For this reason I have only shown Estimated Uniques in the above table to 50 generations. Even at that point I believe the number is well overstated due to the sample size effect.

My database is not a particularly large one, but the growth in relationships is still significant. Furthermore, as the number of known people linked to the ancestral tree for the studied individual grows, the relationship counts grow at a larger but varying rate. These relationships result from the many complex interactions between descendants. Historically, the volume of data is unfortunately weak on the female side. When the data of later generations is examined where full inclusion of female ancestry applies, a much richer set of complex links result. As more people are added to the "tree" and linked to existing ancestors, the Recorded Knowns grow significantly. This has happened over the last five years in my data.

The information I have compiled so far on the ancestry of Prince William shows some interesting numbers. It is a work in progress and details change day by day.

We often hear and see the comment that the royal families are inbred. This is generally a fallacy and derives from an emotive impression of the effects of remote cousin marriages. In the case of Prince William I have analysed his ancestry using my program ATMatch, described below. This shows that in 12 generations of parental ancestry he has a Wright Coefficient of Inbreeding of 0.0052% representing 33 common parental ancestors occupying 113 ancestral positions (out of 16,380). In addition, his Coefficient of Ancestry (a related measure showing where he has common ancestry regardless of which parent has the links) is 1.835% representing 701 ancestral repetitions occupying 2,821 ancestral positions. These coefficients are very insignificant.

My conclusion is that we all probably all have less than 1,000,000 real unique ancestors, within 80 generations.

This conclusion arises from my observation of the data available to me, in particular the first 40 generations. The following indicative curve shows my predicted pattern of incremental unique ancestors for succeeding generations. The vertical axis is the number of uniques per generation, and the horizontal the number of generations. The latter parts of the curve are logically derived, and may ultimately end with two people if taken to the limit.

There are mobility factors in the population that affect groups of people in different ways and alter the ancestral composition accordingly. The land owning classes tend to live and marry within small geographic regions, whereas itinerant workers move around the country so spreading their genes. This seems to be partially offset in the case of royals, as there is considerable mixture across wider social and political boundaries. It will take a lot more investigation before these patterns are clear.

The shape is similar to that of the skewed statistical F Distribution, but I do not profess to have a sensible explanation for this result. The peak appears to occur at around 25 to 30 generations, and moves a bit over the centuries. This may reflect the consequences of wars and famines.

For any individual, a hypothetical set of unique ancestors could accumulate as follows:-

I hope that these notes provoke some further thought and contributions on this topic.

On request I have provided some further details from my study of Prince William's ancestry. Click Here to view the extra information.

My simple FTRIPLET analysis program is freely available here. It is a useful tool to review an individual’s relationships within their databases. It is menu driven and runs under DOS or the command prompt in Windows 95 and NT and reads its data from a GEDCOM file.

Please use the program to examine the fascinating results of the relationships in your own databases, and be sure to let me know if you find any particularly complex or interesting ones. I would appreciate the data to help further research in the theory.

Its features include: -

This Windows program provides a means for comparing two separate ahnentafel lists for the same individual to identify the similarilies and differences between them. I use it to compare the information in my database with that of another person.

Input is through special ahnentafel listings that can be separately derived from a GEDCOM file, or in a specific format from another researcher.

Program features include: -

In addition to my own FTRIPLET and ATMATCH analysis programs, I have made use of a utility program written by Torben B. Andersen. This program is command line driven and provides a number of very useful facilities for examination and manipulation of GEDCOM files. It is called GEDUTIL.

It runs significantly faster than my own program in the calculation of unique ancestors, though providing a different set of other features.

The program was freely available on e-mail request to the author Torben B. Andersen but may no longer be available.

Site Search Facility