  We consider a family of K\"ahler structures on products of 2-spheres, arising
from complex Bott manifolds. These are obtained via iterated $\mathbb
P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We
show that the resulting K\"ahler structures all have identical Chern classes.
We construct Bott diagrams, which are rooted forests with an edge labelling by
positive integers, and show that these classify these K\"ahler structures up to
biholomorphism.
