On the locating chromatic number of infinite trees

The locating chromatic number of a graph is the smallest integer $n$ such that there is a proper $n$-coloring $c$ and every vertex has a unique vector of distances to colors in $c$. We explore the necessary conditions and provide sufficient conditions for an infinite tree to have a finite locating chromatic number. We also give an algorithm for computing the locating coloring of trees that works for both finite and infinite trees.