Can the topological equivalence classes of countable trees always be characterized by a single cardinal invariant, or are there trees whose class size escapes all standard cardinal arithmetic?

Resolving this question would clarify the boundary between combinatorial and set-theoretic phenomena in topological classification problems. If the trichotomy holds in full generality across all set-theoretic universes, it would provide a powerful structure theorem for countable trees analogous to the Laver or Shelah theorems for linear orders. If instead the class sizes are independent of ZFC, it would open a new front connecting tree combinatorics to forcing and inner model theory, and would directly inform the broader program of classifying countable structures up to topological equivalence in descriptive set theory.