I am new to the world of trees and I am trying to make a painless addition to this general definition:
Let $X$ be a topological space and $\mathfrak{T}$ be a collection of sets.
$(\mathfrak{T},\prec)$ is a $tree$ if
(1) $\prec$ is a strict partial order on $\mathfrak{T}$
(2) $\forall x,y\in \mathfrak{T}, \{x\in \mathfrak{T}:x\prec y\}$ is well-ordered by $\prec$
A $tree$ $in$ $X$ is a function $f:\mathfrak{T}\rightarrow X$.
I would now like to add a condition that the elements of $\mathfrak{T}$ are nested, when they are related, with the minimum of machinery.
Basically, I am looking for a graceful transition from the definition of general tree to more specific Cantor set construction, which I will present as a "nested interval tree", still working on the name. Is there a good definition of a nested tree out there that might fit the bill?