Consider an array of variables indexed by the entries of a three
dimensional lattice obeying the relation
f(n+1,i,j)*f(n-1,i,j)=f(n,i-1,j)*f(n,i+1,j)+f(n,i,j-1)*f(n,i,j+1).
This is the octahedron recurrence, which emerged from the study of the
Hirota equation in integrable systems and was brought to the attention of
combinatorialists by Propp. The cube recurrence is a similar relation
introduced by Propp and again defined on a three dimensional lattice. Each
of these recurrences was motivated by the study of the Somos
sequences.

If we fix a roughly two dimensional set of initial conditions, all of the
other f's are rational expressions in the initial terms. Propp conjectured
and Fomin and Zelevinsky proved that in each recurrence these rational
expressions are actually Laurent polynomials. Propp additionally
conjectured that the coefficients of these Laurent polynomials were all
1.

I will describe combinatorial proofs of these conjectures, due to myself
in the octahedral case and joint work between myself and Gabriel Carroll
in the cube case. In the octahedron case, the monomials turn out to be in
bijection with perfect matchings of certain planar graphs, recovering
results of Ciucu and others. In the cube case, the monomials are in
bijection with groves, certain highly symmetric spanning forests deserving
of more study.