  Let $G$ be a simple finite connected graph of order $n$. The detour distance
between two distinct vertices $u$ and $v$ denoted by $D(u,v)$ is the length of
a longest $uv$-path in $G$. A hamiltonian coloring $h$ of a graph $G$ of order
$n$ is a mapping $h : V(G) \rightarrow \{0,1,2,...\}$ such that $D(u,v) +
|h(u)-h(v)| \geq n-1$, for every two distinct vertices $u$ and $v$ of $G$. The
span of $h$, denoted by $span(h)$, is $\max\{|h(u)-h(v)| : u, v \in V(G)\}$.
The hamiltonian chromatic number of $G$ is defined as $hc(G) :=
\min\{span(h)\}$ with minimum taken over all hamiltonian coloring $h$ of $G$.
In this paper, we give an improved lower bound for the hamiltonian chromatic
number of trees and give a necessary and sufficient condition to achieve the
improved lower bound. Using this result, we determine the hamiltonian chromatic
number of two families of trees.
