What is "branchus"

In Greek mythology, the name Branchus refers to the following characters, who may or may not be identical:

1. Branchus, son of Apollo (or son of Smycrus and lover of Apollo); his mother was a woman from Miletus, who, while she was pregnant with him, had a vision of being pierced with a ray of light. Branchus received prophetic abilities from Apollo and introduced the worship of the god at Didyma. He also founded a shrine at Miletus. His descendants, the Branchides, were an influential clan of prophets.

2. Branchus, father of Cercyon, by the nymph Argiope.