One may ask an analogous continuous problem:

Which connected set composed of simple arcs of total length $1$ has the largest convex hull?

If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \sim \frac{1}{6.93}$, whereas a half-circle of length $1$ gives a half-disc of area $\frac{1}{2\pi} \sim \frac{1}{6.28}$, which is larger.

This implies that for large enough $n$, the Steiner tree with one branching point is not optimal.

Edit: I just found that this problem has been solved in the special case that the connected set is a curve; see the following question: Largest convex hull of a unit length path

The three-dimensional case is discussed here: Largest possible volume of the convex hull of a curve of unit length

One may ask an analogous continuous problem:

Which connected set composed of simple arcs of total length $1$ has the largest convex hull?

If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \sim \frac{1}{6.93}$, whereas a half-circle of length $1$ gives a half-disc of area $\frac{1}{2\pi} \sim \frac{1}{6.28}$, which is larger.

This implies that for large enough $n$, the Steiner tree with one branching point is not optimal.

Edit: I just found that this problem has been solved in the special case that the connected set is a curve; see the following question: Largest convex hull of a unit length path

The three-dimensional case is discussed here: Largest possible volume of the convex hull of a curve of unit length

One may ask an analogous continuous problem:

Which connected set composed of simple arcs of total length $1$ has the largest convex hull?

If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \sim \frac{1}{6.93}$, whereas a half-circle of length $1$ gives a half-disc of area $\frac{1}{2\pi} \sim \frac{1}{6.28}$, which is larger.

This implies that for large enough $n$, the Steiner tree with one branching point is not optimal.

One may ask an analogous continuous problem:

Which connected set composed of simple arcs of total length $1$ has the largest convex hull?

If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \sim \frac{1}{6.93}$, whereas a half-circle of length $1$ gives a half-disc of area $\frac{1}{2\pi} \sim \frac{1}{6.28}$, which is larger.

This implies that for large enough $n$, the Steiner tree with one branching point is not optimal.

Edit: I just found that this problem has been solved in the special case that the connected set is a curve; see the following question: Largest convex hull of a unit length path

The three-dimensional case is discussed here: Largest possible volume of the convex hull of a curve of unit length