Refer to the above link to download a Word version of the below
remarks, which might be easier to read.

RL

I finished the second version of the genealogy program referenced in
these threads here: http://tinyurl.com/2ozylv and here: http://tinyurl.com/37avfk
.

Below are the results.

CONCLUSION:

See below for explanation.  The first "N" below is what I term the
"default" or "normal" rate.

N=#boys / Iterations (Simulated Family Histories) / Average Surname
Survival / Extinction Rate

N= 1 or anything / 6000 iterations      / 25%   survival /  75% extinction
rate

N= 2             / 2000 iterations      / 40%   survival /  60% extinction rate

N= 3            / 2000 iterations       / 55%   survival /  75% extinction rate

N= 4             / 1000 iterations      / 67%   survival /  33% extinction rate

N= 5             / 2000 iterations      / 74%   survival /  26% extinction rate

N= 6             / 2000 iterations      / 78%   survival /  22% extinction rate

N= 7             / 1000 iterations      / 82%   survival /  18% extinction rate

N= 8             / 2000 iterations      / 87%   survival /  13% extinction rate

N= 9             / 2000 iterations      / 90%   survival /  10% extinction rate

N= 10            / 3000 iterations      / 92%   survival /  8% extinction rate

BACKGROUND:

The topic is what percent of surnames survive in a population over
time, given that western surname survival is a Galton Watson process
(http://en.wikipedia.org/wiki/Galton-Watson_process).  If you want to
be scientific, it's also the percent survival of the male "Y"
chromosome for any one person or founder of a family line, over time.

The distribution of males that reach reproductive age (hereinafter
"boys") was assumed to be a Poisson distribution having mean (lambda)
1.15, as per a suggestion by WG Whalley, which is the historical
figure for western populations.

Various initial scenarios are assumed.  The normal, and default, is
having any number of boys (including zero), or having one boy in the
first generation (turns out these two scenarios are nearly logically
the same, since having exactly one boy in the first generation simply
shifts the generation survival question by one generation but has no
other effect).  For this first normal scenario of having one or any
other number of boys (including zero) in the first generation, the
survival rate, after 6000 simulations (equivalent to a computer
simulation of a family history) is an average of 25% survival of
surnames.  The other scenarios are having any number of boys N
initially, where N=2,3,4,5,6,7,8,9 and 10; these results are
summarized above.

BIZARRE OR UNUSUAL FINDINGS:

Having a large family with lots of initial boys is no absolute
guarantee of survival of family surname.  While having 9 or 10 boys in
a family gives over 90% survival rate for the surname of the founder
(and/or "Y" chromosome), there's always that 10% chance of going
extinct.  See below examples of families going extinct with 9 or 10
initial boys, in as early as four generations!  This happened quite
often for any run of 1000 simulations.  Four generations is the
shortest I have found for N=9 or 10; nothing even quicker.  For the
below examples of 9 or 10 initial boys that died out in four
generations, you can check for yourself, by hand, that the program
output is correct.

On the other end of the spectrum, the program found many generations
that went extinct even after many previous generations of survival.
This is seen most spectacularly in families with large numbers of
initial boys (because these family surnames usually survive, but when
they don't, it's spectacular to see them fail-and often it takes a
while for these lines to fail).  For example, while most families
either succeed or fail in surname survival before about 25 to 50
generations, a family with two boys failed (surname went extinct) in
156 generations; another family with three boys initially died out
after 88 generations; another family with four initial boys died after
113 generations; another family with five initial boys died after 162
generations; and another with 10 initial boys died in 111 generations!

PROGRAM OUTPUT:

How to read the below array:  the array below shows the offspring of
each male, and is traversed "breath-first" or by "level - order"
traversal in a tree starting from the initial founder (see
http://en.wikipedia.org/wiki/Tree_traversal  note the example "Level-
order traversal sequence: F, B, G, A, D, I, C, E, H"-this is how the
program here works).  The first line is the number of boys the initial
founder had, then the next lines refer to the number of boys his boys
had, and so on, as per the "level-order traversal" example in
Wikipedia, also see this example: http://en.wikipedia.org/wiki/Breadth-first_traversal
.  A simple example:  the lines 1,2,1,2,0,0,0 means the founder had
one boy, who had two boys, each of whom had 1 and 2 boys,
respectively, and these three boys had zero boys (the last three
zeros) so the family surname died out.

Examples (arrays are truncated for brevity):

This array went extinct after 156 generations! (Initial input 2 boys)
---+++------------------
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
Date: 9/30/2007 14:51:28; elap.time 0.2103024 extct? True % :39 itt:
214  cur.gen 156 TheNodeTree.NodeCount() 1434
CandidateNodesQueue.Count:0!END
Time now is9/30/2007 14:51:28
--
This array went extinct after 88 generations (initial input 3 boys)
---+++------------------
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  3
Date: 9/30/2007 14:59:27; elap.time 0.1802592 extct? True % :54 itt:
434  cur.gen 88 TheNodeTree.NodeCount() 784 CandidateNodesQueue.Count:
0!END

This array, with initial input 4 boys, went extinct after 113
generations!
---+++------------------
OffSpringBoys [0,i] is:  4
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
Date: 9/30/2007 15:15:05; elap.time 0.1902736 extct? True % :67 itt:
779  cur.gen 113 TheNodeTree.NodeCount() 1239
CandidateNodesQueue.Count:0!END

--
Generation went extinct despite 10 initial boys (in 111 generations)!
---+++------------------
OffSpringBoys [0,i] is:  10
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
Date: 9/30/2007 15:29:38; elap.time 0.2103024 extct? True % :93 itt:
668  cur.gen 111 TheNodeTree.NodeCount() 1441
CandidateNodesQueue.Count:0!END

--
---+++------------------
--
Generation went extinct despite 10 boys in just 4 gen! (can be
verified below)

---+++------------------
OffSpringBoys [0,i] is:  10
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  3 <--too late, never reached!  Family surname
went extinct just before this

Date: 9/30/2007 15:43:47; elap.time 0.1802592 extct? True % :85 itt:
14  cur.gen 4 TheNodeTree.NodeCount() 19 CandidateNodesQueue.Count:0!
END
--

Went extinct despite 10 boys in a mere 4 generations!
Time now is9/30/2007 16:13:05
---+++------------------
OffSpringBoys [0,i] is:  10
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  3 <--three boys in this line did not help!
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  3 <--three boys in same "line" failed to
help!
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0 <--went extinct here, too late for the 3"in
the next line!
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  3
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  0
OffSpringBoys [0,i] is:  1
OffSpringBoys [0,i] is:  2
OffSpringBoys [0,i] is:  2
Date: 9/30/2007 16:13:05; elap.time 0.2203168 extct? True % :91 itt:
354  cur.gen 4 TheNodeTree.NodeCount() 25 CandidateNodesQueue.Count:0!
END
--

Generation with 9 initial boys that went extinct in four generations!
Time now is10/1/2007 05:23:48

---+++------------------
OffSpringBoys [0,i] is:  9
OffSpringBoys ...

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