Let $G$ be a $k$-uniform hypergraph with vertex set $V(G)$ and edge set $E(G)$. A connected and acyclic hypergraph is called a supertree. For $0\leqα<1$, the $α$-spectral radius of $G$ is the largest $H$-eigenvalue of $αD(G)+(1-α)A(G)$, where $D(G)$ and $A(G)$ are the diagonal tensor of the degrees and the adjacency tensor of $G$, respectively. In this paper, we determine the unique supertrees with the maximum $α$-spectral radius among all $k$-uniform supertrees with $m$ edges and independence number $β$ for $\lceil\frac{m(k-1)+1}{k}\rceil\leqβ\leq m$, among all $k$-uniform supertrees with given degree sequences, and among all $k$-uniform supertrees with $m$ edges and matching number $μ$ for $1\leqμ\leq\lfloor\frac{m(k-1)+1}{k}\rfloor$, respectively.