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.
Since the structure defines an inside and an outside, it might be useful for situation where the nutrients are not absorbed via the roots but via the branches/leaves, while being carried by the medium. More or less what sponges do in water.

Such a tree would therefore have the outer leaves being the photosynthetic ones, while the inner ones would be the absorbing ones.

The roots would be used only as supporting mechanism, to anchor the tree to the substrate.

The environment could, for example, be:

* Highly windy, with nutrients carried by the wind. The hollow shape allows for the carried particles to precipitate.
* Liquid, like for the sponges.