Limit distribution for the maximum degree of a random recursive tree (Q1612294)

Let \(k=\lfloor\ln n/\ln 2\rfloor+ d\) for a fixed integer \(d\). The authors show that the probability that the maximum in-degree of a random recursive tree with \(n\) vertices is at most \(k\) equals \[ \exp(- 2^{\{\ln n/\ln 2\}- d-1})+ o(1) \] as \(n\to\infty\), where \(\{x\}= x-\lfloor x\rfloor\).