Question: Premise
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I was inspired by [Karl Niklas](https://en.wikipedia.org/wiki/Karl_J._Niklas) who used computer modeling to simulate the ideal structure for a tree when different traits were favored. Here is a chart summarizing his work:

[![enter image description here](https://i.stack.imgur.com/llkgX.jpg)](https://i.stack.imgur.com/llkgX.jpg)

**Explanation**

* L (light): horizontal branches
* M (mechanical stability): fewer horizontal branches
* R (reproduction): taller to spread seeds further

When he combined them all, his model yielded trees that closely resemble trees in the real world, with enough of each trait to survive and reproduce.

**A new tree shape**

From this point I wanted to take things to a more extreme angle. I want to have trees that are shaped like a dodecahedron:

[![enter image description here](https://i.stack.imgur.com/xg8ug.png)](https://i.stack.imgur.com/xg8ug.png)

Question
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If we take the point from Niklas' research that trees take a shape that helps them best survive given what the environment deems favorable traits, then what kind of a world do I need for a dodecahedron tree to be deemed favorable?

**Further Clarifications:**

* Roots: as pictured
* Foliage: configurable
* World: configurable (Earth-like or non-Earth like), we just assume no matter what the world, evolutionary theory, as we understand it, holds to a reasonable degree

Answer: 
Trees are remarkably bloody minded in their growth patterns, since their greatest influencing factor is the presence of other trees, the classic requirements indicated in your question override almost anything else.

So we shall not have a tree, instead we shall have a shrub.

Here we get a lot more flexibility as we can spread the driving influences around a little.

We should also talk about [coevolution](https://en.wikipedia.org/wiki/Coevolution). A classic example being bees and flowers. Flowers need bees, bees need flowers. Take away one and the other is lost. The two have coevolved to be largely dependent on each other.

Your pattern of the branches rejoining to form such a shape is highly inefficient and would not happen without some sort of secondary influence driving it that matters more than that inefficiency. What you've created is a distinct inside and outside to the plant, importantly it's an inside that can only be accessed by creatures of a certain maximum size. We shall define it as a coevolved trait.

You have options here:

* flowers and pollination
* seeds and fruit
* protection
* trap

The first is reasonably common, normally it appears in the form of very long flowers matched by hummingbirds with very long beaks or some other similar form.

Fruit are usually designed to be attractive as widely as possible so that they get the best chance of dispersal. Colour schemes are designed to stand out to creatures with colour vision such as birds, monkeys and apes.

Protection could be an interesting one. Some creature finds a safe home in the bush and in doing so protects the bush from other creatures. Acacia Trees and ants sometimes have such a relationship.

It could be a carnivorous tree, the branches and thorns allow small creatures in, but don't allow them out again.

Any or all of the above in combination also works, perhaps in flowering and fruiting season creatures can come and go easily through the branches, but out of season the gaps allow inward movement only.
A permanent tumbling plant may evolve a shape that as closely resembles spherical as possible, while using minimal mass. Natural objects also tends to respect symmetry as this reduces the necessary genome size. Like the icosahedral viruses which only have to code for a protein that forms a pentameter of trimers for its capsid.

I imagine a world with constantly blowing high winds, which will uproot anything that is not adapted against it's force, but will also give any plants that can harness it's force plenty of opportunity to disperse and thrive. For example, plants that can tumble in the wind and put down roots where ever there is plenty of water and nutrient supplies, and have a shape that can always photosynthesize no matter which side it lands on.

For a given shape, the fewer the faces it has, the less amount of edge length it has for a give surface area. Of course, there will also be tetrahedral, cubic and octagonal plants, but they are unable to roll due to having acute to right angles on their edges and vertices.

The plant needs to roll or tumble, so therefore it have to have leaves on all sides, and needs to be rounded for rolling. It also have to be the lightest for its given size. A tree's mass is largely concentrated in its trunk and branches, therefore: any higher fullerenes or spherically symmetric structures will be less mass-efficient for it's cross section, and any lower symmetric structures won't roll because of the angles, therefore a rolling plant optimizes to adopt a dodecahedral shape, to both maximize it's surface area and minimize it's mass, while still being able to roll with the wind.

As for growth, I imagine the tree grows as a trimer of pentamers, I.e. having individual seedlings being a simple bent shape with a bend angle of 108 degrees, every five seedlings locks together to forme a Pentagonal face, then the outside of every three seedlings adheres to each other, forming a vertex. This naturally self assembles into a dodecahedron when high winds jitters the field of newly sprouted seedlings around, causing them to be stired up and bump into each other. As each seedling sprouts roots at the bent point, the dodecahedral structure can always take root nomatter which side it lands on. Also, The leaves will be on the inside of the seedlings, forming the face of the dodecahedron. This ensures the plant always have photosynthetic tissue facing the sun/light source nomatter which face it lands on.