  The avalanche polynomial on a graph captures the distribution of avalanches
in the abelian sandpile model. Studied on trees, this polynomial could be
defined by simply considering the size of the subtrees of the original tree. In
this article, we study some properties of this polynomial on plane trees.
Previously it has been proved that two different trees could have the same
avalanche polynomial. We show here that the problem of finding a tree with a
prescribed polynomial is NP-complete. In a second part, we study the average
and the variance of the avalanche distribution on trees and give a closed
formula.
