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In this paper, we study the joint behaviour of the degree, depth and label ofand graph distance between high-degree vertices in the random recursive tree.We generalise the results obtained by Eslava and extend these to include thelabels of and graph distance between high-degree vertices. The analysis of boththese two properties of high-degree vertices is novel, in particular inrelation to the behaviour of the depth of such vertices. In passing, we alsoobtain results for the joint behaviour of the degree and depth of and graphdistance between any fixed number of vertices with a prescribed label. Thiscombines several isolated results on the degree and depth of and graph distancebetween vertices with a prescribed label already present in the literature.Furthermore, we extend these results to hold jointly for any number of fixedvertices and improve these results by providing more detailed descriptions ofthe distributional limits. Our analysis is based on a correspondence betweenthe random recursive tree and a representation of the Kingman $n$-coalescent.

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