An involution on increasing trees

We introduce a method to construct bijections on increasing trees. Using this method, we construct an involution on increasing trees, from which we obtain the equidistribution of the statistics ‘number of odd vertices’ and ‘number of even vertices at odd levels’. As an application, we deduce that the expected value of the number of even vertices is twice the expected value of the number of odd vertices in a random recursive tree of given size.