The Dying Fibonacci Tree

If p = 1 then the resulting tree is the Fibonacci tree. It can easily be verified that the number of A’s in the n-th layer equals the n-th Fibonacci number Fn and the number of B’s equals Fn−1. Let An and Bn denote the number of A’s and B’s, respectively, in the first n layers of the tree. Then we have An An + Bn = ∑n i=1 Fi ∑n+1 i=2 Fi = 1 − Fn+1 − F1 ∑n+1 i=2 Fi Using the well known representation of the Fibonacci numbers Fn = (α n − α−n)/ √ 5 where α = (1 + √ 5)/2 we immediately get An An + Bn = 1 − α n+1 − α−n−1 − α + 1/α (α2(1 − αn) − α−1(1 − α−n))/(1 − α) ∼ 1 α = √ 5 − 1 2 (1)