  In this paper we introduce a new method called the Dirac Assisted Tree (DAT)
method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily
large variable wave numbers. DAT breaks an original global problem into many
parallel tree-structured small local problems, which are linked together to
form a global solution by solving small linking problems. To solve the local
problems in DAT, we propose a compact finite difference method (FDM) with
arbitrarily high accuracy order and low numerical dispersion for piecewise
smooth coefficients and variable wave numbers. This compact FDM is particularly
appealing for DAT, because the local problems and their fluxes in DAT can be
computed with high accuracy. DAT with such compact FDMs can solve heterogeneous
Helmholtz equations with arbitrarily large variable wave numbers accurately by
solving small linear systems - $4 \times 4$ matrices in the extreme case - with
tridiagonal coefficient matrices in a parallel fashion. Several numerical
examples are provided to illustrate the effectiveness of DAT using the $M$th
order compact FDMs with $M=6,8$ for numerically solving heterogeneous Helmholtz
equations with variable wave numbers. We shall also discuss how to solve some
special 2D Helmholtz equations using DAT.
