*preprint*

**Inserted:** 10 feb 2023

**Last Updated:** 12 feb 2024

**Journal:** Fractal and Fractional

**Volume:** 7

**Number:** 5

**Pages:** 414

**Year:** 2023

**Doi:** https://doi.org/10.3390/fractalfract7050414

**Abstract:**

We consider a general metric Steiner problem which is of finding a set $\mathcal{S}$ with minimal length such that $\mathcal{S} \cup A$ is connected, where $A$ is a given compact subset of a given complete metric space $X$; a solution is called Steiner tree. Paolini, Stepanov and Teplitskaya provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension which was an open question (the corresponding tree is a self-similar fractal).

**Keywords:**
Steiner problem, Steiner tree, fractal set, explicit solution