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The Aldous-Broder algorithm samples a uniform spanning tree of a finite connected graph by running a simple random walk from any initial vertex. Whenever the random walk first visits a new vertex, retain the edge by which it arrived. Stop when every vertex has been visited.

Each retained edge joins a previously unseen vertex to the already visited connected subgraph. It cannot create a graph cycle; at termination the retained edges form a spanning tree. The stopping time is the cover time, finite almost surely. The nontrivial conclusion is that every spanning tree has exactly the same probability, independently of the starting vertex.^[ref]

Repeated visits add no edges, which can make the method slow when only a few unvisited vertices remain. Wilson's algorithm instead attaches loop-erased random walks. On a recurrent infinite connected graph, the first-entrance construction continues indefinitely and gives the corresponding infinite-volume uniform spanning tree.

The Aldous-Broder algorithm starts the recurrent random walk at $x_0$ and, whenever it first visits a vertex $v\ne x_0$, adds the edge by which it entered $v$. Recurrence ensures that every vertex is eventually visited. The collection of first-entrance edges is a spanning tree and has the infinite-volume uniform spanning-tree law.