Trees of definable sets over the $p$-adics

To a definable subset of Znp (or to a scheme of finite type over Zp) one can associate a tree in a natural way. It is known that the corresponding Poincare´ series PNlZl A Z½½Z is rational, where Nl is the number of nodes of the tree at depth l. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z2p , and for one-dimensional sets.