  A weighted recursive tree is an evolving tree in which vertices are assigned
random vertex-weights and new vertices connect to a predecessor with a
probability proportional to its weight. Here, we study the maximum degree and
near-maximum degrees in weighted recursive trees when the vertex-weights are
almost surely bounded and their distribution function satisfies a mild
regularity condition near zero. We are able to specify higher-order corrections
to the first order growth of the maximum degree established in prior work. The
accuracy of the results depends on the behaviour of the weight distribution
near the largest possible value and in certain cases we manage to find the
corrections up to random order. Additionally, we describe the tail distribution
of the maximum degree, the distribution of the number of vertices attaining the
maximum degree, and establish asymptotic normality of the number of vertices
with near-maximum degree. Our analysis extends the results proved for random
recursive trees (where the weights are constant) to the case of random weights.
The main technical result shows that the degrees of several uniformly chosen
vertices are asymptotically independent with explicit error corrections.
