The story
* All three meanings are related to defecation, in three different ways. * Grow comes from the ideas of expansion (after emergence), either sudden (as in senses 1 and 3) or gradual (as in sense 2). * Tail refers to the fecal matter itself, namely its shape (especially that of a so-called “log”, particularly in sense 3) and the side of the body whence it is expeled (the back side, where a tail would have been located).
Etymology from Wiktionary, CC BY-SA 4.0.