The packing and covering number of trees are local

Abstract

For a graph, its $r$-packing number is the size of the largest $r$-separated set and its $r$-covering number is the size of the smallest $r$-net. We show that the $r$-covering number and $r$-packing number are continuous in the fringe topology on locally finite trees. This recovers convergence results by Meir and Moon for e.g. uniform binary trees, uniform plane trees, uniform Polyá trees and random recursive trees, and proves new results for many other tree models, such as size-conditioned Bienaymé trees, simply generated trees, $m$-ary search trees, the minimum spanning tree of the complete graph and preferential attachment trees. Moreover, our formulation provides an intuitive interpretation of the limiting constant, that also provides an easy route for explicit computations in the case of size-conditioned Bienaymé trees. Our method relies on the definition of a canonical minimum $r$-net, and a relation between covering numbers and packing numbers due to Meir and Moon.