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We study the basic preferential attachment process, which generates asequence of random trees, each obtained from the previous one by introducing anew vertex and joining it to one existing vertex, chosen with probabilityproportional to its degree. We investigate the number $D_t(\ell)$ of verticesof each degree $\ell$ at each time $t$, focussing particularly on the casewhere $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ isconcentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell\le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.

Vertices of high degree in the preferential attachment tree