Realization of prescribed distance-shell cardinalities

Determine necessary and sufficient conditions on positive integers \(b_0,\ldots,b_n\) for the existence of a finite ultrametric space \((X,d)\) with center of distances \(C(X)=\{c_0,\ldots,c_n\}\), where \(0=c_0<\cdots<c_n\), and with \(|\{x\in X:d(p,x)=c_i\}|=b_i\) for every \(p\in X\) and every \(i\).

Background

For each point $p$, the quantity $|\{x\in X:d(p,x)=c_i\}|$ records the size of the distance shell at radius $c_i$. The problem asks which finite sequences of shell sizes can occur uniformly at every point of a finite ultrametric space with a prescribed center of distances.

The question is related to the paper’s realization results for finite distance sets and to the subsequent conjecture concerning the special shell sizes of MCD-spaces.