Your question is a lot more complicated than it sounds.
All colours breed predictably according to the genes carried by the parents. (Breed makes absolutely no difference.) In general, every bird has two copies of each gene, and contributes one of each to its offspring. There are a few exceptions; these are the sex-linked genes, in which females carry only one copy. Their daughters inherit the lack of a copy, their sons inherit the copy the mother carries.
For each gene there are two or more alternatives, or alleles. When both copies carried by the bird are the same allele, the bird is said to be homoygous for that gene, and all offspring will inherit that allele as no alternatives are possible from that parent. When a bird carries two different alleles for a gene, the bird is heterozygous for the gene; some of the offspring will inherit one allele, others will inherit the other allele. Each offspring has a 50/50 chance for inheriting each of these alternatives; in large numbers of offspring, about half the chicks will inherit one allele and the other half the alternative allele. In small numbers of chicks, this does not necessarily hold true.
When the bird is heterozygous, the order of dominance among the alleles determine the appearance of the bird. For instance, blue is dominant to not-blue, so a bird carrying blue will display it. Not-lavender is dominant to lavender, so a bird carrying lavender and not-lavender will display not-lavender.
Buff and red are complex; there is more than one "recipe" to create these varieties (buff and red are the same except that buff carries red pigment dilution gene or genes).
Birchen is perhaps an easier answer in that it is based upon a single gene (although all the other genes certainly make their impact). Birchen is the 2nd most dominant of the e-alleles. If the black cochin is homozygous for extended black, the offspring would carry both extended black and birchen, and would display as black. However, melanizers are needed to make the extended black bird completely black, so it is quite possible that the offspring may not inherit sufficient melanizers to be completely black.
If the black parent is black based upon any other e-allele, the birchen will be dominant and show in the offspring.
For fun, go play with Henk's chicken calculator at kippenjungle.nl