  We study the asymptotic behavior of the maximum degree in the evolving tree
model with a choice based on both degree and fitness of a vertex. The tree is
constructed in the following recursive way. Each vertex is assigned a parameter
to it that is called a fitness of a vertex. We start from two vertices and an
edge between them. On each step we consider a sample with repetition of $d$
vertices, chosen with probabilities proportional to their degrees plus some
parameter $\beta>-1$. Then we add a new vertex and draw an edge from it to the
vertex from the sample with the highest product of fitness and degree. We prove
that dependent on parameters of the model, the maximum degree could exhibit
three types of asymptotic behavior: sublinear, linear and of $n/\ln n$ order,
where $n$ is the number of edges in the graph.
