Local extrema in random trees

The extension of permutation statistics to labelled trees is the subject of a number of articles. Generating functions for the number of labelled trees of several types according to the number of ascents and descents are given in [4]. A functional equation satisfied by the generating function for the number of labelled trees according to the number of descents and leaves is given in [5]. Central and local limit theorems for the number of ascents or of descents in uniformly random labelled trees are given in [1]. A functional equation satisfied by the generating function for the number of labelled trees according to the number of inversions is given in [7]. Related results are contained in [6]. A formula for the expected number of inversions of a uniformly random labelled tree is given in [9]. Formulas for the expectation and variance of the number of inversions of a uniformly random labelled tree are given in [2]. Local extrema (in the literature as local maxima and local minima; peaks and troughs; collectively turning points; related to phases) in permutations have a long history; see [10] and references there in. The examination of local maxima (equivalently, local minima) in permutations is more recent. A recurrence relation and a generating function for the number of permutations according to the number of local maxima are given in [10]. A central limit theorem for the number of local maxima in a uniformly random permutation also is given in [10]. In this note, we extend local extrema in permutations to labelled trees and examine local maxima (equivalently, local minima) in uniformly random labelled trees. For n≥ 2, let n denote the set of trees with vertex set [n] := {1, . . . ,n}. When T1,T2 ∈ n, T1 = T2 if and only if T1 and T2 have the same edge set. Let T ∈ n, r ∈ [n], and