Abstract: A sign pattern is a matrix whose entries belong to the set
{+,−, 0}. An n-by-n sign pattern A is said to allow an eventually
positive matrix if there exist some real matrices A with the same
sign pattern as A and a positive integer k0 such that Ak > 0 for all
k ≥ k0. It is well known that identifying and classifying the n-by-n
sign patterns that allow an eventually positive matrix are posed as two
open problems. In this article, the tree sign patterns of small order
that allow an eventually positive matrix are classified completely.