Transplanting a tree across north-south hemispheres?

What would happen if you took a mature deciduous tree in mid September and transplanted it (perfectly, quickly, safely, no trauma) from the United States to a similiar latitutude and climate in the southern Hemisphere? At the time of the transplant, the tree would be transitioning from its summer green leafy season to its leaf shedding season but once it is transplanted it would be relocated to a place where other trees are in Spring and “waking up” to their leafy season. The transplanted tree would be six months out of phase. What would happen? Would it get back in phase? How? How long?