The human face and body are also quasi-symmetrical. Studies have found that higher symmetry tends to be indicative of higher genetic quality, and more symmetrical people are rated as being more beautiful. This would make one think that a perfectly symmetrical person would be the most attractive option possible, but something happens when a face is graphically altered to be perfectly symmetrical, it ends up looking fake and not as attractive. It's as if the imperfections in our symmetry make us human and our quirks can make us somewhat more attractive in some ways.
I find the same is true with trees. A perfectly symmetrical tree does not look natural; not only does it not look natural, but it also tends to be very boring. A tree that is close to being symmetrical, but not quite perfect, tends to look quite attractive and healthy, but still kind of boring. It's the trees that lean to one side or the other, or have a branch missing that are more interesting; they may not be more beautiful, but they tend to be more expressive. This seems to go back to the comparison to humans, a person that is missing a limb or in a pose that distorts their symmetry is much more interesting to look at. A person or tree in a pose that is highly symmetrical tends to suggest stability, as opposed to an asymmetrical pose that suggests instability, change or motion.
So, trees and faces/people have geometrical similarities; the same could be said about many objects. And that brings me to another point, that the human mind tends to see many objects as faces or people; we anthropomorphize things all the time. When you yell at your car because it won't start, you're anthropomorphizing; you're pretending your car can hear you, something it is obviously not capable of. I think in the case of trees, it is especially easy to anthropomorphize because of the geometrical similarities and also the biological similarities; we are both living things that come in slightly different varieties (1) and have 'trunks' and 'limbs.'
When combined, the similarities of trees to people and the socially primed cognitive mechanisms involved in processing such objects may be what leads to the aesthetic and psychosocial primacy afforded to trees over, say, squares.
1. I think this is an important determinant of the anthropomorphizability of an object; if something comes in many slightly differentiated varieties, it may activate cognitive capacities used to differentiate people, who also tend to differ slightly from one another. With this, the object is now being processed as socially salient and our brains may imbue it with a capacity of agency (theory-of-mind attribution). Contrary to this would be squares; you don't really find squares to be all that anthropomorphizable because they are always the same (maybe color or size differences, but mostly the same).
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