Hypoenergetic and strongly hypoenergetic trees

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. An $n$-vertex graph is said to be hypoenergetic if $E(G)<n$ and strongly hypoenergetic if $E(G)<n-1$. In this paper, we consider hypoenergetic and strongly hypoenergetic trees. For any given $n$ and $\mathrm{\Delta }$, the existence of both hypoenergetic and strongly hypoenergetic trees of order $n$ and maximum degree $\mathrm{\Delta }$ is completely characterized.