  An equivalent directed version of the celebrated unresolved conjecture of
Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every
tournament H there exists epsilon(H)>0 such that every H-free n-vertex
tournament T contains a transitive subtournament of order at least
n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0
such that for every H-free tournament T with |T| > 1, there exist disjoint
vertex subsets A and B, each of cardinality at least |T|n and every vertex of A
is adjacent to every vertex of B. Berger et al. proved that the unique
five-vertex tournament denoted by C5, where every vertex has two inneighbors
and two outneighbors has the strong EH-property. It is known that every
tournament with the strong EH-property also has the EH-property. In this paper
we prove that tournaments that can be ordered in a way that the graph formed by
the backedges is a forest consisting of trees with at most two edges and
consecutive leaves under the vertex ordering has the strong EH-property.
