Dirac-type conditions for spanning bounded-degree hypertrees

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n\mathrm{/}2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a sub-graph. This generalises a well-known result of Koml'os, S'ark"ozy and Szemer'edi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.