I then thought of comparing this with a much better documented genealogy of about the same vintage - the descendants of Queen Victoria and Prince Albert. There are apparently 836 of them living at present, which is actually not a bad match for the Bunkers since it covers almost exactly the same time period (their first child was born in 1840). The rate of increase over the intervening period is basically exponential as you would expect:
Converting it to a logarithmic scale gives a better sense of the texture:
After the initial surge as Victorian and Albert produced their 9 children, the average annual increase since the mid-1880s is 2.6% (one can detect perhaps several phases here - a 4% annual rate from 1885 to 1915, then a drop below 2% to 1956, then a sudden surge back up to 3.5% to 1969, and from then on a fairly steady 1.9%). At the 2.6% rate, Victoria and Albert's living descendants will number 1000 by the year 2015, and will reach a million by the year 2300 and a billion by the year 2600. At the recent lower 1.9% rate, it takes longer (1000 by 2018, a million by 2400, a billion by 2800).
I have to admit that Victoria and Albert, on the one hand, and the Bunker brothers, on the other, were pretty philoprogenitive. Shakespeare's direct descendants died out with his granddaughter. More locally, my brother, my sister and I (and our children) are the only living descendants of our American pair of great-grandparents. (Our Irish sets of great-grandparents did rather better.) But I think you quite rapidly reach the point where individual family fortunes get smeared into the overall bigger picture, and once you have over, say, 40 descendants the edge effects become less important.
Of course, this is so far all about descendants through all possible lines. (Also the figures include adopted children, apparently 14 out of the 1000 cases here.) Research into Y-chromosomes and mitochondrial DNA addresses the male and female lines of descent exclusively, so I wondered what these look like for the descendants of Victoria and Albert. This graph shows those living at the end of each year:
As you would expect (because mothers tend to be younger than fathers) the female line is some way above the male line. But the edge effects, ie the fortunes of individual family units, are much more noticeable here. Note in particular that the female-line descendants are consistently around twice as numerous as the male-line descendants, until 1918, when 18% of them (Alexandra and her four daughters) get removed from the equation by firing squad. They are still more numerous than the males, but only by 40-50% these days.
This graph shows the living female-line and male-line descendants as a percentage of all living descendants of Victoria and Albert since 1900. The overall trend for both is down but not out (I reckon both sets of descendants have now reached critical mass). By the time Victoria and Albert have a million descendants, I reckon that only a few hundred will be directly in the male or female lines. This puts the genetic achievements of, say, Genghis Khan (pdf) and Niall of the Nine Hostages into perspective; if 8% of men from China to Uzbekistan are descended from Genghis in the male line, and a similar percentage of Irish men are descended in the male line from Niall, then the entire local population (and much of that of neighbouring or dispora countries) must also be descended from the ancient progenitors one way or the other.
Incidentally, the direct male and female lines from Victoria and Albert do not include the current top rank of British royals. Queen Elizabeth's father, George VI, was a direct male descendant of Victoria and Albert (via George V and Edward VII), but of course had only two daughters so Albert's Y-chromosomes ended with him. Likewise Prince Philip's mother, Princess Alice of Battenberg, was a direct female-line descendant via her mother Princess Victoria of Hesse and her grandmother Princess Alice (so were also Prince Philip's four sisters, now all dead, and their daughters and female-line descendants). One of the more interesting male-line descendants by one of Victoria's younger sons is the Swiss musician Adrian Coburg, an expert in Afro-Cuban percussion.
Going back to where I started, a couple who lived in the ninth century, and whose descendants consistently increased by 2% per annum since then without other constraints, would now have descendants equal in number to the population of the whole world today. If the growth rate is 1.5%, you have to kick the most recent common ancestor at constant growth rates back to the fifth century. Of course geographical and cultural constraints will slow the process down; but I think Rohde showed convincingly that it makes surprisingly little difference to the basic conclusion, which is that we are probably all descended from someone who lived in south-eastern China, who had a lot of children, and who died less than two thousand years ago.