GIT: the difference between HEAD~1 and HEAD^1

Source: Stackoverflow --> git-rev-parse manual page

^N: Nth parent of the tip of the current branch.
~N: Nth generation first parent of the tip of the current branch.

Here is an illustration, by Jon Loeliger. Both commit nodes B and C are parents of commit node A. Parent commits are ordered left-to-right.

G   H   I   J
 \ /     \ /
  D   E   F
   \  |  / \
    \ | /   |
     \|/    |
      B     C
       \   /
        \ /
         A
A =      = A^0
B = A^   = A^1     = A~1
C = A^2  = A^2
D = A^^  = A^1^1   = A~2
E = B^2  = A^^2
F = B^3  = A^^3
G = A^^^ = A^1^1^1 = A~3
H = D^2  = B^^2    = A^^^2  = A~2^2
I = F^   = B^3^    = A^^3^
J = F^2  = B^3^2   = A^^3^2

(emphasis by me)