  Consider the following partial "sorting algorithm" on permutations: take the
first entry of the permutation in one-line notation and insert it into the
position of its own value. Continue until the first entry is 1. This process
imposes a forest structure on the set of all permutations of size $n$, where
the roots are the permutations starting with 1 and the leaves are derangements.
Viewing the process in the opposite direction towards the leaves, one picks a
fixed point and moves it to the beginning. Despite its simplicity, this "fixed
point forest" exhibits a rich structure. In this paper, we consider the fixed
point forest in the limit $n\to \infty$ and show using Stein's method that at a
random permutation the local structure weakly converges to a tree defined in
terms of independent Poisson point processes. We also show that the
distribution of the length of the longest path to a leaf converges to the
geometric distribution with mean $e-1$, and the length of the shortest path
converges to the Poisson distribution with mean 1. In addition, the higher
moments are bounded and hence the expectations converge as well.
