In, On and At

"In the corner": is used when referring to a concept of closed spaced.
Example- The chair is in the classroom.

"On the corner" or "At the corner": are used to indicate an open space, say street or park.
Sentences:
The tree is at the corner of the street
The tree is on the corner of the street

My assumption is that if the tree is planted right on the corner, we use "on". If it is round the corner, we use "at".

"On time" and "In time"

On time: at the planned time; neither late nor early.
Example- Peter wants the meeting to start exactly on time.

"In time": with enough time to spare; before the last moment.
Example- He would have died if they hadn´t got him to the hospital in time.