  Maxmin trees are labeled trees with the property that each vertex is either a
local maximum or a local minimum. Such trees were originally introduced by
Postnikov, who gave a formula to count them and different combinatorial
interpretations for their number. In this paper we generalize this construction
and define tiered trees by allowing more than two classes of vertices. Tiered
trees arise naturally when counting the absolutely indecomposable
representations of certain quivers, and also when one enumerates torus orbits
on certain homogeneous varieties. We define a notion of weight for tiered trees
and prove bijections between various weight 0 tiered trees and other
combinatorial objects; in particular order n weight 0 maxmin trees are
naturally in bijection with permutations on n-1 letters. We conclude by using
our weight function to define a new q-analogue of the Eulerian numbers.
