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On a connected locally finite recurrent graph, root Wilson's algorithm at a fixed graph vertex $r$ and process a countable enumeration of the graph vertices. Every walk hits the existing tree almost surely, since it contains $r$. Each attached graph path is finite, and the union is a connected acyclic spanning subgraph. Its law is the infinite uniform spanning tree.

It is also the common free and wired finite-volume limit of a sequence along every graph exhaustion. For a finite edge set, run a finite initial segment of the chosen enumeration containing all its endpoints. Its membership is then permanently decided, because later attachments have new internal graph vertices. Only finitely many finite random-walk trajectories were used, and their graph vertices and graph neighbours fit inside all sufficiently large exhaustion sets. Coupling then gives convergence of those finite-dimensional edge laws for both boundary conventions. Finite Wilson's algorithm is root/order independent, so the limiting law is too. There is no uniform counting measure on all infinite spanning trees; the definition is this limiting probability law.

On a finite connected unweighted graph, the graph path between $x$ and $y$ in a uniform spanning tree has the law of loop-erased random walk from $x$ to $y$. Root Wilson's algorithm at $y$ and process $x$ first: the first attached graph path is exactly that loop erasure, and later attachments cannot change it. Changing the root to $x$ proves reversibility of loop-erased random walk, even though chronological loop erasure does not commute with reversing each individual random-walk trajectory.