In a one-parameter model for evolution of random trees, which also includes the Barabasi-Albert random tree , almost sure behavior and the limiting distribution of the degree of a vertex in a fixed position are examined. A functional central limit theorem is also given. Results about Polya urn models are applied in the proofs.
Agnes Backhausz. "Limit distribution of degrees in random family trees." Electron. Commun. Probab. 16 29 - 37, 2011. https://doi.org/10.1214/ECP.v16-1598