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The closest arrangement of trees in which they are at least a certain distance away from each other is a triangular lattice, much like a honeycomb. I wonder if you tend to see this sort of arrangement of trees in the wild?
I guess this would actually not be beneficial to the trees since it would mean they would all be lined up in certain directions which might not be good for sunlight etc.
I wonder if there is much research on the pattern of distribution of trees in a forest?
For most trees and plants, distribution is spatially highly variable. Thinking about the lifecycle of plants—their seeds are non-randomly dispersed, the seedlings and saplings grow in different biotic (neighboring trees, distance to herbivores) and abiotic (different soils, shading, etc.) environments, all the plants are at different life/size stages, and that they are highly plastic in their ability to have fluid morphologies in different conditions—it's really difficult to detect a clean lattice pattern with little variance.
Nevertheless, for some species of plants, where the conditions are right (e.g., very limited and highly sought-after resources, species poor areas, synchronicity of generations, strong antagonism), we see some evidence of regular patterns. Ecologists often analyze spatial distributions to see if they are random, clumped, or uniform/regularly distributed (link). We see some distributional uniformity that does resemble lattice-/tiling-like patterns, which you may find in the references from the link above. There is reference to only 1 tree species in that link :/ I can't say that I see ecologists looking for lattices explicitly, but I would be surprised if there isn't a subfield interested in it. HTH.
It would be closer to an irregular Voronoi pattern.
"A certain distance" is imprecise, it would have to be "regular" intervals to give a lattice/matrix.
For trees that grow at the same rate, a square grid arrangement would give square canopy shapes. A triangle lattice would give triangles and honeycomb would give honeycomb canopy units.
For trees that are different sizes, the regularity would be chaotic. Voronoi: 
Voronoi can describe a lot of things in nature, bubbles and animal cells, it's one of the fundamental mathematical patterns in nature just like spirals. 
And even some arid clay. 
Trees have irregular, not straight sides, that's different from voronoi. The trees are spaced irregularly and chaotically, and so it's a chaos of differently sized and irregular cells, rather than a lattics.
