  Maxmin trees are trees that consist of nodes that are either local minimums
or maximums. Such trees were first studied by Postnikov. Later Dugan, Glennon,
Gunnells, and Steingrimsson introduced the concept of weight to these trees and
proved a bijection between maximum weight maxmin trees and permutations,
defining weights for permutations. In addition, the q-Eulerian polynomial
$E_n(x, q)$ is defined which relates descents and weights of permutations. This
polynomial was later proven to exhibit a stabilization phenomenon by Agrawal et
al. Extracting the formal power series $W_d(t)$ from the stabilization of these
coefficients, $W_d(t)$ was conjectured to partially correspond to A256193. In
our paper, we introduce a process called minimum decomposition to help us
better understand maxmin trees. Using minimum decomposition, we present a new
way to calculate the weight of different maxmin trees and prove the bijection
between the coefficients of $W_d(t)$ and A256193.
