There are two basic types of branches of zoology: Specialization by types of animals, and specialization by processes or functions of animals.
Specializations across types of animals may be broad categorizations of animals, or may become very specialized to just a few or even one species. For example, entomologists specialize in insects. There are thousands upon thousands of types of insects, but what they all have in commons is that they are members of the arthropods and have exactly six legs and four wings. Some entomologists have more narrow specializations, such as those who study roaches, mosquitoes, or the social insects.
Other broader categorizations, such as herpetologists who study snakes, have significantly fewer species to study. Primatologists, who study primates, have fewer still.
So branches of zoology who study certain kinds of animals tend to specialize by animal type. They may be more generalized or more specific, depending on their research interests. There is a virtually endless list of specializations, when you consider the number of Phyla in existence past and present, the number of classes, orders and families, and the species that comprise them.
The other type of branching in zoology would have to do with specializing in certain processes or functions. Examples of this type of specialization would be those who study biodiversity, comparative anatomy, ecosystems, and even breeding programs. Specialists here have to have a broader set of learning backgrounds that would include related fields.
For example, those who study biodiversity, in addition to having their training in zoology, would also have to have a firm foundation in ecology, botany, and systems theory. Those who study breeding programs would have to be well versed in certain medical fields such as nutrition, reproductive systems, and so on.
The number of specializations by processes or functions continues to grow in science as well. As a natural result of the unfolding of scientific understanding, specializations arise in terms of research needs.
If contemplating a career or educational pursuit in zoology, one should research the programs offered at institutions, and learn what kinds of working conditions are suitable. Zoologists may be strictly laboratory in the nature of their work, or may be almost completely in the field, or some combination of the two.