On joint properties of vertices with a given degree or label in the random recursive tree

Abstract

In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava [12] and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree [22], depth [7,24] and graph distance [9,15] of vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree known and a representation of the Kingman n-coalescent.