A note on subtrees rooted along the primary path of a binary tree

Let F_n denote the set of rooted binary plane trees with n external nodes, for given T∈F_n let u_i(T) be the altitude i node along the primary path of T, and let δ_i(T) denote the number of external nodes in the induced subtree rooted at u_i(T). We set δ_i(T) = 0 if i is greater than the length of the primary path of T. We prove lim_n→∞ ∑_i≤x/n E_n{δ_i}/∑_i<∞ E_n{δ_i} = G(x), where E_n denotes the average over trees T∈F_n and where the distribution function G is determined by its moments, for which we present an explicit expression.