  We compute the Euler characteristics of quiver Grassmannians and quiver flag
varieties of tree and band modules and prove their positivity. This generalizes
some results by G.C. Irelli [arXiv:0910.2592].
  As an application we consider the Ringel-Hall algebra $C(A)$ of some string
algebras $A$ and compute in combinatorial terms the products of arbitrary
functions in $C(A)$.
