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Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w\left(e\right) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \inE$ have a "distance" $d \left(e, f\right) = d \left(f, e\right) \in \mathbb{R}$assigned to them, such that the distances satisfy the ultrametric triangleinequality $d(a,b)\leqslant \max \left\{d(a,c),d(b,c)\right\}$. We look for asubset of $E$ of given size with maximum perimeter (where the perimeter isdefined by summing the weights of all elements and their pairwise distances).We show that any such subset can be found by a greedy algorithm (which startswith the empty set, and then adds new elements one by one, maximizing theperimeter at each step). We use this to define numerical invariants, and alsoto show that the maximum-perimeter subsets of all sizes form a strong greedoid,and the maximum-perimeter subsets of any given size are the bases of a matroid.This essentially generalizes the "$P$-orderings" constructed by Bhargava inorder to define his generalized factorials, and is also similar to the stronggreedoid of maximum diversity subsets in phylogenetic trees studied by Moulton,Semple and Steel.  We further discuss some numerical invariants of $E, w, d$ stemming from thisconstruction, as well as an analogue where maximum-perimeter subsets arereplaced by maximum-perimeter tuples (i.e., elements can appear multipletimes).

A greedoid and a matroid inspired by Bhargava's $p$-orderings