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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?

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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.

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    $\begingroup$ Good answer. I was surprised that there are actually plants that have regular distributions. I wonder though whether they are categorised as separate plants or are they in fact a giant super organism with a common root structure. In any case it seems like some form of communication between the different out-growths is needed for this arrangement. Also, I would imagine that such arrangement must grow out from a single plant much like how a snowflake crystal grows from a single seed. $\endgroup$ – zooby May 15 at 21:20
  • $\begingroup$ Thanks—good question. Some plants that tend to grow regularly distributed, like creosote (Larrea tridentata) some oaks, can reproduce clonally and through seeds. Clonal propagation ought to lead to more regular distributions. Communication in most plants through root exudates are ubiquitous and very poorly understood, but I think you're right on about stronger patterns arising from sensing and responding to local environments. Animals do this well, and many territories (again, in the right conditions) are lattice-like, such as nesting sites of birds that next at high densities in the open. $\endgroup$ – Chris Moore May 16 at 16:46
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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: enter image description here enter image description here

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. enter image description here And even some arid clay. enter image description here 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. enter image description here

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  • $\begingroup$ I think it would not quite be veronoi because there's only so close trees can get. And trees would tend to overshadow their neighbours and kill them. $\endgroup$ – zooby May 15 at 17:41
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    $\begingroup$ I think you have things backwards. You can make a Voroni diagram for ANY distribution of points, since it's merely the area that's closer to each point than any of the others: en.wikipedia.org/wiki/Voronoi_diagram $\endgroup$ – jamesqf May 15 at 18:05
  • $\begingroup$ @zooby You are correct that the points of any distribution can be joined togeter into triangles... Google "Voronoi Forest" you will find lots of images and mathematics forestry research using voronoi... revistas.inia.es/index.php/fs/article/view/8021/2913 lattice is a repeating arrangement, i.e it has a symmetry. mathematics research has used voronoi to define plant spaces: pdfs.semanticscholar.org/de5a/… $\endgroup$ – com.prehensible May 15 at 20:23
  • $\begingroup$ OK, then what's the point in saying it's veronoi? Why not just say it's an irregular distribution of points? I guess you are trying to show how points can form cell like patterns? $\endgroup$ – zooby May 15 at 21:17
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    $\begingroup$ @zooby: Re "only so close trees can get", not really. It's not uncommon to find two trees growing so close together that their trunks meet and sometimes join: en.wikipedia.org/wiki/Inosculation $\endgroup$ – jamesqf May 16 at 16:47

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