Definable B-sigma-two path

Construct a definable path h through a Pi-zero-one(emptyset-prime) tree T such that M[h] satisfies B-Sigma-two, assuming that some path g through T has this property.

Background

This is the tree version of the preceding definable-choice question. The issue is whether the model-theoretic property can be preserved while replacing an arbitrary witnessing path by a definable one.