The bandwidth of a tree with k leaves is at most [k2]

The bandwidth B(G) of a finite simple graph G is the minimum of the quantity max{ƒ(x)-ƒ(y) : xy ∈ E(G)} taken over all injective integer labellings ƒ of G. We prove that if a tree T has k leaves then B(T) ⩽ [k2]. This improves the previously known upper bound B(T) ⩽ ∥ V(T)∥/2.