A note on isoperimetric peaks of complete trees

We prove that the edge isoperimetric peak of the complete k -ary tree of height h is Θ ( h k ) . For the vertex isoperimetric peak we prove a value of Θ ( h ) . Both results improve on the previous best lower bounds of Ω ( h k ) and Ω ( h / k ) , respectively, from [B.V. Subramanya Bharadwaja, L. Sunil Chandran, Bounds on isoperimetric values of trees, Discrete Mathematics 309 (2009) 834–842].