  For a finite-dimensional Frobenius $k$-algebra $R$ with the Nakayama
automorphism $\nu$ we define an algebra ${\rm HH}^*(R)^{\nu\uparrow}$. If the
order of $\nu$ is not divisible by the characteristic of $k$, this algebra is
isomorphic to the Hochschild cohomology algebra of $R$.. We prove that this
algebra is a BV-algebra. We use this fact to calculate the Gerstenhaber algebra
structure and BV-structure on the Hochschild cohomology algebras of a family of
self-injective algebras of tree type $D_n$.
