The height of depth-weighted random recursive trees

In this paper, we introduce a model of depth-weighted random recursive trees, created by recursively joining a new leaf to an existing vertex v. In this model, the probability of choosing v depends on its depth in the tree. In particular, we assume that there is a function f such that if k has depth k then its probability of being chosen is proportional to f(k). We consider the expected value of the diameter of this model as determined by f, and for various increasing f we find expectations that range from polylogarithmic to linear.