Motivation
I have been trying to digest the paper "Fast Construction of Nets in Low Dimensional Metrics, and Their Applications", which has relevance to many problems in computational topology and geometry, but have been having trouble gaining an intuition for the authors' definition of a net-tree.
This concept is of central importance to the results of the paper, but despite mulling it over for some hours I can't seem to grasp it.
Background
The general idea behind the paper, as I understand it, is to provide a data structure that approximates a metric space but in a more computationally efficient way.
A well-established notion of a "sparse" approximation of a metric space is given by (what the authors call) an $r$-net, defined as follows:
An $r$-net in a metric space $\mathcal{M}$ is a subset $\mathcal{N} \subset \mathcal{M}$ of points such that $\sup _{x \in \mathcal{M}} d_{\mathcal{M}}(x, \mathcal{N}) \leq r$ and $\inf _{x, y \in \mathcal{N} ; x \neq y} d_{\mathcal{M}}(x, y) \geq r / \alpha$ for some constant $\alpha \geq 1$.
The authors proceed to define the notion of a net-tree, which I understand to be a "multiscale" analog of an $r$-net. The definition is given as follows. [Note that $\mathbf{b}(x,r)$ denotes the closed ball of radius $r$ centered at $x$]
Let $P \subset \mathcal{M}$ be a finite subset. A net-tree of $P$ is a tree $T$ whose set of leaves is $P$. We denote by $P_v \subset P$ the set of leaves in the subtree rooted at a vertex $v \in T$. Associate with each vertex $v$ a point $\operatorname{rep}_v \in P_v$. Internal vertices have at least two children. Each vertex $v$ has a level $\ell(v) \in \mathbb{Z} \cup \{ -\infty \} $. The levels satisfy $\ell(v)<\ell(\overline{\mathrm{p}}(v))$, where $\overline{\mathrm{p}}(v)$ is the parent of $v$ in $T$. The levels of the leaves are $-\infty$. Let $\tau$ be some large enough constant, say $\tau=11$.
We require the following properties from $T$ :
Covering property: For every vertex $v \in T$, $$ \mathbf{b}\left(\operatorname{rep}_v, \frac{2 \tau}{\tau-1} \cdot \tau^{\ell(v)}\right) \supset P_v $$
Packing property: For every nonroot vertex $v \in T$, $$ \mathbf{b}\left(\operatorname{rep}_v, \frac{\tau-5}{2(\tau-1)} \cdot \tau^{\ell(\overline{\mathrm{p}}(v))-1}\right) \bigcap P \subset P_v . $$
Inheritance property: For every nonleaf vertex $u \in T$, there exists a child $v \in T$ of $u$ such that $\operatorname{rep}_u=\operatorname{rep}_v$.
Question
Is there a geometric (or otherwise intuitive) interpretation of the above definition?
At first glance, it seems rather arbitrary and opaque to me. In particular, it isn't clear to me why one wants these properties, or these specific definitions of them.
I did not find reading further into the paper to be helpful in terms of elucidating this concept, but perhaps someone who is more familiar with these ideas can express the definition in a more intuitive way.