  We consider random partitions of the vertex set of a given finite graph that
can be sampled by means of loop-erased random walks stopped at a random
independent exponential time of parameter $q>0$. The related random blocks tend
to cluster nodes visited by the random walk associated to the graph on time
scale $1/q$. This random partitioning is induced by a measure of spanning
rooted forest of the graph which naturally generalize the classical uniform
spanning tree measure and which can be obtained as a "zero-limit" of
FK-percolation with an external cemetery state. General properties of this
rooted forest measure and related determinantal observables, along with a
number of applications in data analysis have been recently explored. We are
here interested in understanding the emergent partitioning, referred to as
loop-erased partitioning, as the intensity parameter $q$ scales with the graph
size. Some first results in this direction have been investigated in the recent
[arXiv:1906.03858] for dense geometries. In this work we instead look at very
sparse geometries. We start by characterising monotone events in $q$ by
deriving a general Russo-like formula for this rooted arboreal gas measure. We
then explore the resulting clusters on simple insightful tree-like graphs by
looking at the probability that two given vertices do not belong to the same
block. In particular, a detailed analysis of the resulting integer partitioning
on line segments is pursued. We finally look at simple trees and other almost
tree-like geometries, without and with implanted modular structures. For the
latter, we characterize the emergence of giants and asymptotic detection of
these implanted modules as the scale parameter $q$ varies.
