  We study the local limit of the fixed-point forest, a tree structure
associated to a simple sorting algorithm on permutations. This local limit can
be viewed as an infinite random tree that can be constructed from a Poisson
point process configuration on $[0,1]^\mathbb{N}$. We generalize this random
tree, and compute the expected size and expected number of leaves of a random
rooted subtree in the generalized version. We also obtain bounds on the
variance of the size.
