Let T be a homogeneous tree of homogeneity q+1. Let Δ denote the boundary of T, consisting of all infinite geodesics b=[b0,b1,b2,] beginning at the root, 0. For each bεΔ, τ≥1, and a≥0 we define the approach region Ωτ,a(b) to be the set of all vertices t such that, for some j, t is a descendant of bj and the geodesic distance of t to bj is at most (τ−1)j+a. If τ>1, we view these as tangential approach regions to b with degree of tangency τ. We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition ∑Tfp(t)q−γ|t|<∞, where p>1 and 0<γ<1, or p=1 and 0<γ≤1. For 1≤τ≤1/γ, we show that Gf(s) has limit zero as s approaches a boundary point b within Ωτ,a(b) except for a subset E of Δ of τγ-dimensional Hausdorff measure 0, where Hτγ(E)=sup δ>0inf ∑iq−τγ|ti|:E a subset of the boundary points passing through ti for some i,|ti|>log q(1/δ).