I figured out the problem. For others encountering the same thing, here's the solution:
The filename ("TreeTest.png") needs to go at the end of the mogrify command to function properly. From man page:
mogrify [options] input-file
This command works as expected:
mogrify -resize 144x216 -background none -gravity South -extent 192x288 TreeTest.png
(The only difference is 'TreeTest.png' was moved to the end of the command)