  Let $\Delta(d,n)$ denote the maximum diameter of a $d$-dimensional polyhedron
with $n$ facets. In this paper, we propose a unified analysis of a recursive
inequality about $\Delta(d,n)$ established by Kalai and Kleitman in 1992. This
yields much simpler proofs of a tail--polynomial and tail--almost--linear
bounds on $\Delta(d,n)$ which are recently discussed by Gallagher and Kim.
