As the title implies, the essay's thesis is how the city is not a tree. To compare trees and cities, the author presents a rather abstract (but accessible) analysis based on semi-lattices.
Alexander's argument is quite convincing but the essay's end is what really got me:
"For the human mind, the tree is the easiest vehicle for complex thoughts. But the city is not, cannot and must not be a tree. The city is a receptacle for life. If the receptacle severs the overlap of the strands of life within it, because it is a tree, it will be like a bowl full of razor blades on edge, ready to cut up whatever is entrusted to it. In such a receptacle life will be cut to pieces. If we make cities which are trees, they will cut our life within to pieces."
I tried to draw this out but I forgot the key sentence!
I highly recommend reading this essay (only a couple of pages long WITH awesome figures. The essay can be read here: https://www.patternlanguage.com/archive/cityisnotatree.html