Usually the start of a river is called its source and the end of the river is called the mouth.
A stream or river called that flows into a larger stream or river is called a tributary.
The source
Tributary is the answer you are looking for, I believe.
it forms at the beginning of a stream or river
A small stream in a river is called a current.
The source or beginning of a river is called the 'headwater'. The beginning of a river is known as the source.
The middle part of a river may be called mid-stream, mid-river or mature river.
Are you crazy? A stream or a small river is called a TRIBUTARY.
A tributary.
A stream?
A stream.
River