Let Γ be a torsion-free cocompact lattice in Aut(T₁) × Aut(T₂), where T₁, T₂ are trees whose vertices all have degree at least three. The group H₂(Γ, Z) is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product C*-algebra A associated with the action of Γ on the boundary of T₁×T₂ satisfies rank K₀(A) = 2⋅rank H₂(Γ, Z).