Do trees in a forest generally grow in a lattice formation?
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.