The white tree shoot is a quest item used in the Garden of Tranquility quest. The player must speak to the gardener Dantaera just east of Catherby with activated Ring of Charos equipped. She will tell the player that the White tree is dying and the only way to save it is to grow a cutting from one of its branches.
Currently, players could cut from the Ice Mountain white tree even after completing the quest. They could even plant it in a plant pot and water it, but it would never grow into a sapling, and cannot be planted into a wood or fruit tree patch, so it would just be a waste of plant pots.