Metric dimension of critical Galton–Watson trees and linear preferential attachment trees

The metric dimension of a graph G is the minimal size subset R vertices that, upon reporting their distance from distinguished (source) vertex v?, enable unique identification source v? among all possible G. In this paper we show Law Large Numbers (LLN) for some classes trees: critical Galton–Watson trees conditioned to have n, and growing general linear preferential attachment trees. former class includes uniform random trees, latter Yule-trees (also called recursive trees), m-ary increasing binary search positive these cases, are able identify limiting constant in LLN explicitly. Our result relies on insight that can be related subtree properties, hence make use powerful fringe-tree literature developed by Aldous Janson et al.