  We study a general model of recursive trees where vertices are equipped with
independent weights and at each time-step a vertex is sampled with probability
proportional to its fitness function (a function of its weight and degree) and
connects to $\ell$ new-coming vertices. Under a certain technical assumption,
applying the theory of Crump-Mode-Jagers branching processes, we derive
formulas for the almost sure limiting distribution of the proportion of
vertices with a given degree and weight, and proportion of edges with endpoint
having a certain weight. As an application of this theorem, we prove rigorously
observations of Bianconi related to the evolving Cayley tree in [$\mathit{Phys.
\, Rev. \, E} \; \mathbf{66}, \text{ 036116 (2002)}$]. We also study the
process in depth when the technical condition can fail in the particular case
when the fitness function is affine, a model we call "generalised preferential
attachment with fitness". We show that this model can exhibit condensation
where a positive proportion of edges accumulate around vertices with maximal
weight, or, more drastically, have a degenerate limiting degree distribution
where the entire proportion of edges accumulate around these vertices. Finally,
we prove stochastic convergence for the degree distribution under a different
assumption of a strong law of large numbers for the partition function
associated with the process.
