The topical rain forest has trees which shed leaves at different times of the year. It is not like all the trees shed their leaves at the same time of the year. Different trees shed leaves at different times of the yea which means while few trees are shedding leaves, others are not. This means that a certain portion of the forest is still green. For example, if the forest has 100 A trees and 100 B trees. Let's say during January A trees shed their leaves. During this period B trees are not shedding their leaves. Which means a portion of the forest is still green. In March lets say B Trees are shedding leaves. A trees have just got new leaves and so that part of the forest is green. Now the forest has trees in random. It doesn't have A trees on one side and B trees on another side. Which means due to random placement of the trees in the forest, the forest will always remain green and hence the name Evergreen forest.

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The basic logic behind the name of evergreen forest is that, these forests are hot throughout the year and also get rainfall throughout the year. That too, heavy rainfall. So, different tree species have adapted to live in this condition maintaining the ecology and biodiversity of the area.
So, different trees shed leaves at different times of the year.
So, the forest appears green throughout the year. Hence, the name. A tree May become leaf less for sometime, but the new leaves grow soon. Leaves are not lost altogether in this forest.
Hope this helps.
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