Invisibi;ity Cloak

[snapesmate]

No: HPFGUIDX 73738

 I haved looked but cannot find any old posts about this, so 
forgive me if I am re-hashing an old subject.  I wondered what your 
theories are as to who gave the cloak to Harry and why?  I have 
wondered if it was Snape.  I do not know why James would leave the 
cloak with Snape, but... In PoA, Snape sneaks into the wamping willow 
using the cloak and after revealing himself even comments that he 
used "Potter's" invisibility cloak.  How did he know it was Harry's?  
I know there was the Hogsmeade incident when Malfoy said he saw 
Potter's head, LOL, but even then no one mentioned that they 
suspected the rest of Harry was hidden by the cloak. As I  said, what 
do you all think?