Proximity to ${\mathrm{\ell }}_p$ and $c_0$ in Banach spaces

We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain ${\mathrm{\ell }}_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain ${\mathrm{\ell }}_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X\mathrm{/}Y$.