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Let $G$ be a large group acting on a biregular tree $T$ and $\Gamma \leq G$ ageometrically finite lattice. In an earlier work, the authors classified orbitclosures of the action of the horospherical subgroups on $G/\Gamma$. In thisarticle we show that there is no escape of mass and use this to prove that, infact, dense orbits equidistribute to the Haar measure on $G/\Gamma$. On theother hand, we show that new dynamical phenomena for horospherical actionsappear on quotients by non-geometrically finite lattices: we give examples ofnon-geometrically finite lattices where an escape of mass phenomenon occurs andwhere the orbital averages along a Folner sequence do not converge. In the lastpart, as a by-product of our methods, we show that projections to $\Gamma\backslash T$ of the uniform distributions on large spheres in the tree $T$converge to a natural probability measure on $\Gamma \backslash T$. Finally, weapply this equidistribution result to a lattice point counting problem toobtain counting asymptotics with exponential error term.

(Non)-escape of mass and equidistribution for horospherical actions on trees