  We construct a stationary random tree, embedded in the upper half plane, with
prescribed offspring distribution and whose vertices are the atoms of a unit
Poisson point process. This process which we call Hammersley's tree process
extends the usual Hammersley's line process. Just as Hammersley's process is
related to the problem of the longest increasing subsequence, this model also
has a combinatorial interpretation: it counts the number of heaps (i.e.
increasing trees) required to store a random permutation. This problem was
initially considered by Byers et. al (2011) and Istrate and Bonchis (2015) in
the case of regular trees. We show, in particular, that the number of heaps
grows logarithmically with the size of the permutation.
