The Pessimistic Search and the Straightening Involution for Trees

We introduce the idea of pessimistic search on a rooted tree, and develop the straightening involution to relate the inversion polynomial evaluated atq = − 1 to the number of even rooted trees. We obtain a differential equation for the inversion polynomial of cyclic trees evaluated atq = − 1, a problem proposed by Gessel, Sagan and Yeh. Some brief discussions about relevant topics are also presented.