  A hamiltonian coloring $c$ of a graph $G$ of order $n$ is a mapping $c$ :
$V(G) \rightarrow \{0,1,2,...\}$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$
$n-1$, for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$
denotes the detour distance between $u$ and $v$ which is the length of a
longest $u,v$-path in $G$. The value $hc(c)$ of a hamiltonian coloring $c$ is
the maximum color assigned to a vertex of $G$. The hamiltonian chromatic
number, denoted by $hc(G)$, is the min{$hc(c)$} taken over all hamiltonian
coloring $c$ of $G$. In this paper, we present a lower bound for the
hamiltonian chromatic number of trees and give a sufficient condition to
achieve this lower bound. Using this condition we determine the hamiltonian
chromatic number of symmetric trees, firecracker trees and a special class of
caterpillars.
