We show that $(n,d,λ)$-graphs with $λ=O(d/\log^3 n)$ are universal with respect to all bounded degree spanning trees. This significantly improves upon the previous best bound due to Han and Yang of the form $λ=d/\exp{(O(\sqrt{\log n}))}$, and makes progress towards a problem of Alon, Krivelevich, and Sudakov from 2007. Our proof relies on the existence of sorting networks of logarithmic depth, as given by a celebrated construction of Ajtai, Komlós and Szemerédi. Using this construction, we show that the classical vertex-disjoint paths problem can be solved for a set of vertices fixed in advance.