Increasing trees and alternating permutations

In this article we consider some increasing trees, the number of which is equal to the number of alternating (updown) permutations, that is, permutations of the form σ(1) < σ(2) > σ(3) < ... . It turns out that there are several such classes of increasing trees, each of which is interesting in itself. Special attention is paid to the study of various statistics on these trees, connected with the Andre polynomials and the Foata group on the one hand, and the Entringer numbers, which arise on grading alternating permutations from the first element, on the other. The proofs of most of the assertions are based on the construction of explicit bijections between classes of trees and the study of their properties. We also obtain new combinatorial identities for the Euler and Bernoulli numbers. In §2 we give basic definitions and known theorems concerned with alternating permutations, binary and increasing trees. We define 0-1-2 trees, that is, trees such that at most two edges go out from any vertex, we establish a bijection between them and orbits of the action of the Foata group on binary trees, and consider various statistics. In §3 we define even trees, that is, trees such that an even number of edges go out from each non-rooted vertex. We find an explicit bijection between them and alternating permutations, and establish a connection with the inversion polynomial and the Tutte dichromate of a complete graph.