  Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. The main purpose
of this paper is to explore the question of uniform sampling of a subtree of
$G$ with $n$ nodes, for some $n\leq N$ (the spanning tree case correspond to
$n=N$, and is already deeply studied in the literature). We provide new
asymptotically exact simulation methods using Markov chains for general
connected graphs $G$, and any $n\leq N$. We highlight the case of the uniform
subtree of $\mathbb{Z}^2$ with $n$ nodes, containing the origin $(0,0)$ for
which Schramm asked several questions. We produce pictures, statistics, and
some conjectures. A second aim of the paper is devoted to surveying other
models of random subtrees of a graph, among them, DLA models, the first passage
percolation, the uniform spanning tree and the minimum spanning tree. We also
provide new models, some statistics, and some conjectures.

Probability