Suppose you moved a deciduous tree (say a maple) from Vermont to the tropics (on the equator). Would this tree drop its leaves (as it would do in the Northern Hemisphere)? How would it sense the change of seasons? Or would it keep its leaves and stay green all year?
By definition, a deciduous tree would lose its leaves. But you may not be drawing the distinction correctly.
Elms, oaks, maples, ashes. poplars, rowans, etc., are “broadleaf” trees – their leaves are flat-surfaced (whether single or compound). Virtually all are dicotyledenous angiosperms. Contrast this to confiers, all gymnosperms of the Conferaceae=Pinaceae (I think I have this right) whose leaves are needles or scales (pine, spruce, yew, fir, cedar, juniper).
Deciduous trees shed their leaves in fall, en masse. Evergreen trees, on the other hand, lose theirs a few at a time on an ongoing basis, replacements growing in as the old leaves are shed.
In the North of the U.S. and northern/central Europe, nearly all broadleaf trees are deciduous. Further south, magnolia, holly, and other broadleaf trees are evergreen, though most remain deciduous. And of course nearly all conifers are evergreen. (I believe there are a couple of deciduous conifers, though I’m drawing a blank on specifics.) The point is that deciduous/evergreen and broadleaf/conifer are sepaate dichotomies, not quite identical in which trees they describe.
Even in areas where subfreezing temperatures are rare, deciduous trees have a brief dormancy. The same or closely related species begin leafing out here in North Carolina a month to six weeks ahead of what they did in Upstate New York when we lived there, and lose their leaves later in the fall – but they do go through a shorter time of leafless dependency.
So the answer is, if you got a temperate-zone deciduous tree to grow in a tropical area, it would go through a leaf-fall, dormancy, leaf-out period just as in Boston – but it would most likely be a much shorter period than a Boston winter.
I had some personal experience with a deciduous broadleaf tree in California where I grew up. I don’t know the type, but everyone referred to it as a tree that would prefer a colder climate and was confused by the mild winters. Around mid-November, it would start turning orange and lose a few of its leaves each week throughout the winter. In February, it had maybe 30% of the leaves left… and it would hurry up and lose the rest of those even as its new leaves were already budding out.
It depends entirely on the mechanism by which the tree senses seasons.
Most temperate deciduous trees are daylength sensitive. When the number of hours of daylight declines below X hours they shed their leaves. They regrow them when daylength rises above some other threshold. Take those trees to the tropics and they may never shed their leaves because the daylength never falls below the threshold. That is usually fatal to the tree after a few years. Other species may have a sufficiently log daylength trigger that they do shed their leaves but, as Polycarp notes, the period of time between the daylength indicators may be a period of weeks, so the tree might not even finish shedding leaves before producing new leaves.
Some trees are responsive to signals other than absolute daylength. Some respond to relative daylength, so they shed 8 months after the solstice or 6 months after the equinox. They have no problems whether grown in the tropics or temperate regions because those dates are the same everywhere. Others respond to factors such as temperature or soil moisture. These are the most common triggers for tropical deciduous plants though it also applies to some temperate species. These trees may or may not have problems if relocated depending on the precise mechanism.
BTW, you are aware that there is just as much tropical and temperate area in the northern hemisphere as the south? The comparison is between the tropics and the temperate region, not between Northern and Southern Hemispheres.