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.