Not sure how to write the following

Nope.

Nuh-uh.
"I can see the shadows of the moving in the wind trees." <--- Maybe. Or "I can see the shadows of the moving through the wind trees." It's odd to be using "moving" as a noun.

What on earth is "the moving"? You can see movement, and you can also see moving leaves, but you cannot see "THE moving". And there's no such thing as "wind trees" unless it's a fictional type of trees, in which case, by all means keep it. But if "wind trees" was used as a play on language or some kind of poetry, then it doesn't work because I've no idea what "wind trees" are meant to be.

The only way I can understand the sentence is this: "I can see the shadows of the rustling (or moving) leaves in the trees" or "The wind touched the leaves, swaying them in its wake" or something.

There are no trees made of wind. Maybe the shades are drawn in the forest.