Bill wrote:
Here's a nice problem (an example of a "Galton-Watson" process). Start
with 1 thing "alive" at generation 0.
Assume it has a 25% chance of dying, a 50% chance of living, and a 25%
chance of doubling after each generation.
Assume this is true of all such "things". What is the probability
that there will be exactly 1 thing alive after 2 generations?

I believe that a great solution technique to problems like this has
been (re-)discovered numerous times.
Hint: If the question is changed to What is the probability that there
will be exactly k things alive after n generations?
The answer is the same as the value of the coefficient on x^k of the
function f(x)=(1/4 + 1/2 x + 1/4 x^2) composed with itself n times.
That this is true I find pretty darn amazing. And it follows from the
Binomial formula, which you brought up.
The books I've seen leave the reader to figure that out for
themselves, so I won't take the fun out of it.
Suggestion: Start with a "probability tree".


You can probably see how to use this idea to help estimate the
distribution of the population of trees n years from now, if you plant a
new one today.
Of course, there are "overcrowding" issues, but you may be okay for
small values of n. So it is on topic. ; )
Any answers for the question given? I've been working on a related one
all afternoon so it is fresh in my mind.