  With the help of various square principles, we obtain results concerning the
consistency strength of several statements about trees containing ascent paths,
special trees, and strong chain conditions. Building on a result that shows
that Todor\v{c}evi\'{c}'s principle $\square(\kappa)$ implies an indexed
version of $\square(\kappa,\lambda)$, we show that for all infinite, regular
cardinals $\lambda<\kappa$, the principle $\square(\kappa)$ implies the
existence of a $\kappa$-Aronszajn tree containing a $\lambda$-ascent path. We
then provide a complete picture of the consistency strengths of statements
relating the interactions of trees with ascent paths and special trees. As a
part of this analysis, we construct a model of set theory in which
$\aleph_2$-Aronszajn trees exist and all such trees contain $\aleph_0$-ascent
paths. Finally, we use our techniques to show that the assumption that the
$\kappa$-Knaster property is countably productive and the assumption that every
$\kappa$-Knaster partial order is $\kappa$-stationarily layered both imply the
failure of $\square(\kappa)$.
