  A remarkable result of Stanley shows that the set of maximal chains in the
non-crossing partition lattice of type $A$ is Schur-positive, where descents
are defined by a distinguished edge labeling. A bijection between these chains
and labeled trees was presented by Goulden and Yong. Using Adin-Roichman's
variant of Bj\"orner's $EL$-labeling, we show that the subset of maximal chains
in the non-crossing partition lattice of type $A$, whose underlying tree is a
convex caterpillar, is Schur-positive.
