  In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a
dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct
subset of vertices from $C$, is called an identifying code of $G$. The concept
of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in
1998. For a given identifiable graph $G$, let $\M(G)$ be the minimum
cardinality of an identifying code in $G$. In this paper, we show that for any
connected identifiable triangle-free graph $G$ on $n$ vertices having maximum
degree $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. This bound is
asymptotically tight up to constants due to various classes of graphs including
$(\Delta-1)$-ary trees, which are known to have their minimum identifying code
of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for
restricted subfamilies of triangle-free graphs, and conjecture that there
exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}{\Delta}+c$
holds for any nontrivial connected identifiable graph $G$.
