  We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and
of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree
of bounded vertex degree as a subgraph. 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.
