Distribution of pinned distance trees in the plane Fp2

The recent result of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set A⊂R2, if the Hausdorff dimension of A is greater than 54, then the distance set Δ(A) has positive Lebesgue measure. With a completely different approach, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for E⊂Fp2 with |E|≫p5/4, there exist many points x∈E such that the number of distinct distances from x is at least cp. The main purpose of this paper is to prove extensions in the more general structure of pinned trees. The case of distances on small sets will also be addressed in this paper.