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.
The other answers are great, so I'll go a slightly different direction: imagine a [carnivorous plant](https://en.wikipedia.org/wiki/Carnivorous_plant) which folds and unfolds (you can [see an animation of a folding dodecahedron here](https://www.youtube.com/watch?v=Pcbjholy55A)).

If the center edge which stays fixed to the ground (with the roots) has some type of substance to stick or trap prey (or if they folding is fast and reflexive enough), then folding around it as this shape would be a strong force down (making it difficult for prey to escape), and would aid in the digestion process. As an aside, a plant that can eat a lion is creepy.

The challenge is to explain - why a dodecahedron? Why not a circle? Complex shapes do occur in nature, but usually only at very, very small sizes, not at all in plant-and-animal sizes. Perhaps a planet did have a more simple shaped plant or shrub, but as @cyber101 mentions in his answer, over time a shape which prevents grazing, has more structural force, etc, might slowly evolve.

Point is, anything trapping prey in a way which requires the entire inner area to be closed-off with force would inevitably require symmetry. The specifics of why this shape could be happenstance as to the nature of the planet and the creatures around it. I don't see why this wouldn't be *possible* in some theoretical planet, although I admit it would be extremely unlikely.