This is the bike my father gave me as my birthday present.

This is the bike my father gave me as my birthday present.
This is the bike my father gave me as my birthday present.这是我父亲送给我作为生日礼物的单车.
这是一句定语从句吧?
我觉得the bike在定语从句里应该是做宾语吧?那gave 后面是不是应该加个to,才构成my father gave the bike to me这个从句呢?
如果没有to,是my father gave me the bike吗?那the bike是做宾语还是宾语补足语?