  In this work we investigate into energy complexity, a Boolean function
measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the
standard basis $\{\vee_2,\wedge_2,\neg\}$, the energy complexity of
$\mathcal{C}$, denoted by $\mathrm{EC}(\mathcal{C})$, is the maximum number of
its activated inner gates over all inputs. The energy complexity of a Boolean
function $f$, denoted by $\mathrm{EC}(f)$, is the minimum of
$\mathrm{EC}(\mathcal{C})$ over all circuits $\mathcal{C}$ computing $f$. This
concept has attracted lots of attention in literature. Recently, Dinesh, Otiv,
and Sarma [COCOON'18] gave $\mathrm{EC}(f)$ an upper bound in terms of the
decision tree complexity, $\mathrm{EC}(f)=O(\mathrm{D}(f)^3)$. They also showed
that $\mathrm{EC}(f)\leq 3n-1$, where $n$ is the input size. Recall that the
minimum size of circuit to compute $f$ could be as large as $2^n/n$. We improve
their upper bounds by showing that
$\mathrm{EC}(f)\leq\min\{\frac12\mathrm{D}(f)^2+O(\mathrm{D}(f)),n+2\mathrm{D}(f)-2\}$.
For the lower bound, Dinesh, Otiv, and Sarma defined positive sensitivity, a
complexity measure denoted by $\mathrm{psens}(f)$, and showed that
$\mathrm{EC}(f)\ge\frac{1}{3}\mathrm{psens}(f)$. They asked whether
$\mathrm{EC}(f)$ can also be lower bounded by a polynomial of $\mathrm{D}(f)$.
In this paper we affirm it by proving
$\mathrm{EC}(f)=\Omega(\sqrt{\mathrm{D}(f)})$. For non-degenerated functions
with input size $n$, we give another lower bound
$\mathrm{EC}(f)=\Omega(\log{n})$. All these three lower bounds are incomparable
to each other. Besides, we also examine the energy complexity of $\mathtt{OR}$
functions and $\mathtt{ADDRESS}$ functions, which implies the tightness of our
two lower bounds respectively. In addition, the former one answers another open
question asking for a non-trivial lower bounds for the energy complexity of
$\mathtt{OR}$ functions.
