# Family tree numbering

When you draw a tree of your ancestors, things quickly get out of hand. There are twice as many nodes each time you go back a generation, and so the size of paper you need grows exponentially. Things also get messy because typically you know much more about some lines than others. If you know much about your ancestry, one big tree isn’t going to work.

There’s a simple solution to this problem, one commonly used in genealogy: assign everyone in the tree a number, starting with yourself as 1. Then follow two simple rules:

1. The father of person n has number 2n.
2. The mother of person n has number 2n + 1.

You can tell where someone fits into the tree easily from their number. Men have even numbers, women odd numbers. The number of someone’s child is half their number (rounding down if you get a fraction). For example, person 75 on your tree must be a woman. Her husband would be 74, her child 37, her father 150, etc.

Taking the logarithm base 2 tells you how many generations back someone is. That is, person n is ⌊ log2n ⌋ generations back. Going back to our example of 75, this person would be 6 generations back because log2 75 = 6.2288. (Here ⌊ x ⌋ is the “floor” of x, the largest integer less than x. Using the same notation, the child of n is ⌊ n/2 ⌋.)

Said another way, the people m generations back have numbers 2m through 2m+1 – 1. Your paternal line has numbers equal to powers of 2, and your maternal line has numbers one less than powers of 2.

If you write out a person’s number in binary, you stick a 0 on the end to find their father and a 1 on the end to find their mother. So your paternal grandmother, for example, would have number 101 in binary. Start with your number: 1. Then tack on a zero for your father: 10. Then tack on a 1 for his mother: 101.

In our example of 75 above, this number is 1001011 in binary. Leave off the one on the left, then read from left to right saying “father” every time you see a 0 and “mother” every time you see a 1. So person 75 is your father’s father’s mother’s father’s mother’s mother.

This numbering system goes back to at least 1590. In that year Michaël Eytzinger published the chart in the image above, giving the genealogy of Henry III of France.

# Comedic genealogy

Austin Kleon on imitation and originality:

Johnny Carson tried to be Jack Benny but ended up Johnny Carson. David Letterman tried to copy Johnny Carson but ended up David Letterman. And Conan O’Brien tried to be David Letterman but ended up Conan O’Brien. In O’Brien’s words, “It is our failure to become our perceived ideal that ultimately defines us and makes us unique.”

Stolen from Steal Like an Artist

# Mathematical genealogy

The Mathematics Genealogy Project keeps track mathematics PhD students and advisors. I was surprised to find that such information has been preserved for hundreds of years. I was able to trace my mathematical lineage back to Marin Mersenne (1588–1648) of Mersenne prime fame.

I did my PhD under Ralph Showalter, who studied under Tsuan Wu Ting, and so on back to Siméon Poisson (1781–1840).

Then things start to become more complicated. Poisson had two advisors: Joseph Louis Lagrange and Pierre-Simon Laplace. Lagrange also had two advisors: Leonhard Euler and Johann Bernoulli. Etc. One line goes back to Mersenne. Another line goes back to Demetrios Kydones (1324–1397).

Update: Thanks to Frederik Hermans for creating a graph by crawling the Mathematics Genealogy Project site and using Graphviz. It’s too big to view as an ordinary image; the graph gets very bushy in the 16th century. Here’s a PDF version that lets you zoom in and out to see the whole thing.

I was surprised to see Erasmus on the graph. I didn’t run across him when I was just clicking around the website.

# Honeybee genealogy

Male honeybees are born from unfertilized eggs. Female honeybees are born from fertilized eggs. Therefore males have only a mother, but females have both a mother and a father.

Take a male honeybee and graph his ancestors. Let B(n) be the number of bees at the nth level of the family tree. At the first level of the tree is our male honeybee all by himself, so B(1) = 1. At the next level of our tree is his mother, all by herself, so B(2) = 1.

Pick one of the bees at level n of the tree. If this bee is male, he has a mother at level n+1, and a grandmother and grandfather at level n+2. If this bee is female, she has a mother and father at level n+1, and one grandfather and two grandmothers at level n+2. In either case, the number of grandparents is one more than the number of parents. Therefore B(n) + B(n+1) = B(n+2).

To summarize, B(1) = B(2) = 1, and B(n) + B(n+1) = B(n+2). These are the initial conditions and recurrence relation that define the Fibonacci numbers. Therefore the number of bees at level n of the tree equals F(n), the nth Fibonacci number.

This is a more realistic demonstration of Fibonacci numbers in nature than the oft-repeated rabbit problem.