Local extrema in random trees

The number of local maxima (resp., local minima) in a tree T∈n rooted at r∈[n] is denoted by Mr(T) (resp., by mr(T)). We find exact formulas as rational functions of n for the expectation andvariance of M1(T) and mn(T) when T∈n is chosenrandomly according to a uniform distribution. As a consequence,a.a.s. M1(T) and mn(T) belong to a relatively small intervalwhen T∈n