The Dementors were known to not only suck the happiness out of one's soul, but to suck it out of any living thing. What is a tree's "happiness"? Assuming they did not possess sentience, then the soul could be interpreted as heat ...
In past centuries, it was supposed that cold could be reflected, like heat can. Science-wise, we know that cold is not a property in itself, but an absence of heat. So the chill the Dementors left behind was the absence of happiness itself. And it was so cold that the humid air around the trees froze. This may be the truest expression of despair and sadness, indeed!
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