Universal asymptotic parameters for quadratic branching families

Determine whether there are other natural parameters, besides the number of red vertices, whose asymptotic averages are equal across all classes of rooted trees with quadratic branching functions.

Background

The paper proves that the average number of red vertices in trees with ideals is asymptotically n/3 for every considered tree family whose root decomposition is governed by a quadratic branching function. It contrasts this universality with the average number of blue vertices in antichains, whose limiting proportion depends on the branching function. The authors explicitly ask whether analogous universal asymptotic behavior occurs for other natural tree parameters.