How does this look like a tree?

Question

In a book there is a definition like that:

Let $\mathcal{T}$ be the family of subsets $T$ of $S=\bigcup_{n\geq 1}\mathbb{N}^n$ such that $\sigma \mathord{\upharpoonright} k\in T$ whenever $\sigma\in T$ and $1\leq k\leq \#(\sigma)$. Menbers of $\mathcal{T}$ are often called trees.

An example I made: $\{(1), (1, 2), (1, 2, 3), (1, 3), (1, 3, 5), (5), (5, 4), (5, 4, 3)\}$.

How does this look like a tree? Is that a tree in the graph-theoretic sense of being connected and having no closed path?

Answer 1

Does it look like a tree?

Maybe.