Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal{T}$of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal{T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell_p$, then for all $\varepsilon>0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.