  We find representation type of the cyclotomic quiver Hecke algebras of level
two in affine type A. In particular, we have determined representation type for
all the block algebras of Hecke algebras of classical type (except for
characteristic two in type D), which has not been known for a long time. As an
application of this result, we prove that block algebras of finite
representation type are Brauer tree algebras whose Brauer trees are straight
lines without exceptional vertex if the Hecke algebras are of classical type
and the characteristic of the base field is odd. We conjecture that this
statement should hold for Hecke algebras of exceptional type with bad primes
invertible in the base field.
