Enumerative Results On Labeled Trees And Relevant Identities

Trees are one of the most important conceptions in combinatorics and graph theory. As also as a very significant data structure extensively used in computer science and bioin-formatics. Rooted labeled trees are one of the most vital research contents in enumerative combinatorics as a fundamental combinatorial structure. They are closely associated with alternating permutations, hyperplane and other structures. In this paper we give a survey on the recent enumerative results of some labeled trees and related combinatorial identities. We give the counting results of CDG trees, LCM trees and ordered alternating trees as well as related identities. We also study general alternating trees with both generating function method and combinatorial method. Different from classic methods and explanations in enu-merative combinatorics, we concentrate on the diverse combinatorial proofs and extensions of these labeled trees.