Local extrema in random trees

The number of local maxima (resp., local minima) in a tree T ∈ 𝒯 n rooted at r ∈ [ n ] is denoted by M r ( T ) (resp., by m r ( T ) ). We find exact formulas as rational functions of n for the expectation and variance of M 1 (T) and m n ( T ) when T ∈ 𝒯 n is chosen randomly according to a uniform distribution. As a consequence, a.a.s. M 1 ( T ) and m n ( T ) belong to a relatively small interval when T ∈ 𝒯 n .