  In this paper, we completely characterize the digraphs of order $n$ whose
$m$-step competition graphs are star graphs for positive integers $2\leq m <
n$. This result in matrix version identifies the solution set to the matrix
equation $X^m(X^T)^m= \Lambda_n+I_n$ for positive integers $2\leq m < n$ where
$I_n$ is the identity matrix of order $n$ and $\Lambda_n$ is a $(0,1)$ Boolean
matrix such that the first row and the first column consist of $1$'s except
$(1,1)$-entry and the remaining entries are $0$, which is the adjacency matrix
of a star graph of order $n$.
  We also derive meaningful properties of the digraphs whose $m$-step
competition graphs are trees. In the process, we extend a result of
Helleloid~[Connected triangle-free $m$-step competition graphs, Discrete Appl.\
Math.\ 145 (2005) 376--383] by showing that for all positive integers $m \geq
2$ and $n$, the connected triangle-free $m$-step competition graph on $n$
vertices is a tree.
