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We prove the existence of harmonic functions f on trees, with respect to suitable transient transition operators P, that satisfy an analogue of Menshov universal property in the following sense: f is the Poisson transform of a martingale on the boundary of the tree (equipped with the harmonic measure m induced by P) that, for every measurable function h on the boundary, contains a subsequence that converges to h in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.

Frequently dense harmonic functions and universal martingales on trees