Enumeration results for alternating tree families

We study two enumeration problems for up–down alternating trees  , i.e., rooted labelled trees TT, where the labels v1,v2,v3,…v1,v2,v3,… on every path starting at the root of TT satisfy v1v3⋯v1v3⋯. First we consider various tree families of interest in combinatorics (such as unordered, ordered, dd-ary and Motzkin trees) and study the number TnTn of different up–down alternating labelled trees of size nn. We obtain for all tree families considered an implicit characterization of the exponential generating function T(z)T(z) leading to asymptotic results of the coefficients TnTn for various tree families. Second we consider the particular family of up–down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters label of the root node, degree of the root node and depth of a random node   in a random tree of size nn. This leads to exact enumeration results and limiting distribution results.