  We discuss how the Gross-Siebert reconstruction theorem applies to the local
mirror symmetry of Chiang, Klemm, Yau and Zaslow. The reconstruction theorem
associates to certain combinatorial data a degeneration of (log) Calabi-Yau
varieties. While in this case most of the subtleties of the construction are
absent, an important normalization condition already introduces rich geometry.
This condition guarantees the parameters of the construction are canonical
coordinates in the sense of mirror symmetry. The normalization condition is
also related to a count of holomorphic disks and cylinders, as conjectured in
our work and partially proved in various works of Chan, Cho, Lau, Leung and
Tseng. We sketch a possible alternative proof of these counts via logarithmic
Gromov-Witten theory.
  There is also a surprisingly simple interpretation via rooted trees marked by
monomials, which points to an underlying rich algebraic structure both in the
relevant period integrals and the counting of holomorphic disks.
