  Let $\C_n$ be the set of Dyck paths of length $n$. In this paper, by a new
automorphism of ordered trees, we prove that the statistic `number of exterior
pairs', introduced by A. Denise and R. Simion, on the set $\C_n$ is
equidistributed with the statistic `number of up steps at height $h$ with
$h\equiv 0$ (mod 3)'. Moreover, for $m\ge 3$, we prove that the two statistics
`number of up steps at height $h$ with $h\equiv 0$ (mod $m$)' and `number of up
steps at height $h$ with $h\equiv m-1$ (mod $m$)' on the set $\C_n$ are `almost
equidistributed'. Both results are proved combinatorially.
