On Nodes of Given Out-Degree in Random Trees

Let F denote a simply generated family of rooted trees whose generating function y(x) satisfies the relation y - xθ(y) ; suppose that τθ′(τ) = θ′(τ) and that certain technical conditions are met. Let H(j,n) denote the expected height of the j th endnode of a tree selected at random from the trees in F with n nodes (where the endnots are labelled from left to right). The author shows that if j, n → ∞ in such a way that v = j/n is fixed and νθ(τ) H(j,n) ∼ λn 1/2 where λ = 8θ′(τ). {v(1 - vθ(τ))/2πθ″(τ)} 1/2