  We give exact values and bounds on the isoperimetric peak of complete trees,
improving on known results. For the complete $q$-ary tree of depth $d$, if
$q\ge 5$, then we find that the isoperimetric peak equals $d$, completing an
open problem. In the case that $q$ is 3 or 4, we determine the value up to
three values, and in the case $q=2$, up to a logarithmic additive factor. Our
proofs use novel compression techniques, including left, down, and aeolian
compressions. We apply our results to show that the vertex separation number
and the isoperimetric peak of a graph may be arbitrarily far apart as a
function of the order of the graph and give new bounds on the pathwidth and
pursuit-evasion parameters on complete trees.
