Optimal node ranking of trees (Q1113677)

We discuss the problem of ranking nodes of a tree, which is a restriction of the general node coloring problem. A tree is said to have rank number k if its vertices can be ranked using the integers 1,2,...,k such that if two nodes have the same rank i, then there is a node with rank greater than i on the path between the two nodes. The optimal rank number of a tree gives the minimum height of its node separator tree. We present an O(n log n) algorithm for optimal node ranking of trees.