Brindle is a mutation of the gene that causes dominant black as stated in the link that I posted to the Schmutz site. "All of the following dog breeds are black, brown or grey because of KB and could be tested for homozygosity. Both the brindle mutation and the ky allele are recessive to KB. Distinguishing these two recessive alleles with a "simple" DNA test is not yet possible since the brindle mutation is a complex mutation."
Recessive Reds cannot produce black hairs so cannot be brindle.
The genetic research done by the canine genome project supports that phenotypically similar traits found in different breeds are most often caused by the same genes. In this case we have phenotypically similar traits in the same breed, so though without DNA testing it can't be definitively proven, it is more likely than not that they are the same genes. They definately behave the same, lines with barred tris produced recessive blacks in both the pre-Avongara cases and now in the Avongara one.