First, let us consider this simple problem. Assume you a number n and a fraction (between 0 and 1) p. How many times do you need to multiply n with p so that resulting number is less than or equal to 1?
n*p^k <= 1
log(n)+k*log(p) <= 0
log(n) <= -k*log(p)
k => -log(n)/log(p)
Now, let us consider your problem. Assume you send the shorter of the two segments to the left child and longer to the right child. For the left-most chain, the length is given by substituting \alpha as p in the above equation. For the right most chain, the length is calculated by substituting 1-\alpha as p. Which is why you have those numbers as answers.