Ascents and descents in random trees

Abstract The number of descents (respectively, ascents) in a tree T ∈Tn rooted at r∈[n] is denoted d r (T) (respectively, a r (T)) where Tn is the set of all trees with vertex set [n]. Let d r (n, k) (respectively, a r (n, k)) denote the number of trees in Tn rooted at r with precisely k descents (respectively, k ascents). Among our results are central limit theorems and local limit theorems for d r (T) and a r (T) for r=1, n when T is chosen uniformly at random from Tn. We also give asymptotic formulas for max k d r (n, k) and max k a r (n, k) and formulas for their modes for r=1, n.