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 Todorčević's principle (κ) implies an indexed version of (κ, λ), we show that for all infinite, regular cardinals λ < κ, the principle (κ) implies the existence of a κ-Aronszajn tree containing a λ-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 ℵ 2 -Aronszajn trees exist and all such trees contain ℵ 0 -ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of (κ).2010 Mathematics Subject Classification. Primary 03E05; Secondary 03E35, 03E55.