A distance metric based on tree paths

Question

Let $T$ be a connected and acyclic 1-dimensional subset of the plane (a union of curves). For every two points $ְA,B$ in $T$, there is a unique simple path between $A$ and $B$ along the lines of $T$. As an example, here:

T is represented by the black solid lines, and the unique path between A and B is represented by the green dotted line.

Therefore, there is a well-defined distance $d(A,B)$, which is the shortest distance that we have to travel, along lines of $T$, to get from $A$ to $B$. This $d$ is metric, and I am studying its properties. Is there a term that describes this metric, by which I can search for such properties? I tried "tree distance" but it lead to entirely different concepts.

Answer 1

The type of space you're looking at is a real tree:

https://en.wikipedia.org/wiki/Real_tree

Answer 2

In general given a path connected metric space $X$, we can define it's intrinsic metric to be

$$d_I(x, y) = \inf_{p \text{ a path from $x$ to $y$}} \ell(p) $$ where $\ell(p)$ denotes the length of the path.