Small step-dominating sets in trees

We prove that for every tree T=(V,E)T=(V,E) of diameter D⩾3D⩾3 there is a set S⊆VS⊆V with |S|=D-1|S|=D-1 and a mapping st:S→Nst:S→N such that for every vertex v∈Vv∈V there is exactly one vertex u∈Su∈S whose distance to vv equals st(u)st(u). This settles a conjecture of Dror et al. [Some results in step domination of trees, Discrete Math. 289 (2004) 137–144].