  The energy of a molecular graph is a popular parameter that is defined as the
sum of the absolute values of a graph's eigenvalues. It is well known that the
energy is related to the matching polynomial and thus also to the Hosoya index
via a certain Coulson integral. Trees minimizing the energy under various
additional conditions have been determined in the past, e.g., trees with a
given diameter or trees with a perfect matching. However, it is quite a natural
problem to minimize the energy of trees with bounded maximum degree--clearly,
the case of maximum degree 4 (so-called chemical trees) is the most important
one. We will show that the trees with given maximum degree that minimize the
energy are the same that have been shown previously to minimize the Hosoya
index and maximize the Merrifield-Simmons index, thus also proving a conjecture
due to Fischermann et al. Finally, we show that the minimal energy grows
linearly with the size of the trees, with explicitly computable growth
constants that only depend on the maximum degree.
