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We prove the following indistinguishability theorem for $k$-tuples of treesin the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ isa property of a $k$-tuple of components that is stable under finitemodifications of the forest. Then either every $k$-tuple of distinct trees hasproperty $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees hasproperty $\mathscr{A}$ almost surely. This generalizes the indistinguishabilitytheorem of the author and Nachmias (2016), which applied to individual trees.Our results apply more generally to any graph that has the Liouville propertyand for which every component of the USF is one-ended.

Indistinguishability of collections of trees in the uniform spanning forest