Models of random subtrees of a graph

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≤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≤N. We highlight the case of the uniform subtree of Z2 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.