  We study two related probabilistic models of permutations and trees biased by
their number of descents. Here, a descent in a permutation $\sigma$ is a pair
of consecutive elements $\sigma(i), \sigma(i+1)$ such that $\sigma(i) >
\sigma(i+1)$. Likewise, a descent in a rooted tree with labelled vertices is a
pair of a parent vertex and a child such that the label of the parent is
greater than the label of the child. For some nonnegative real number $q$, we
consider the probability measures on permutations and on rooted labelled trees
of a given size where each permutation or tree is chosen with a probability
proportional to $q^{\text{number of descents}}$. In particular, we determine
the asymptotic distribution of the first elements of permutations under this
model. Different phases can be observed based on how $q$ depends on the number
of elements $n$ in our permutations. The results on permutations then allow us
to characterize the local limit of descent-biased rooted labelled trees.
