  A $k$-plane tree is a plane tree whose vertices are assigned labels between
$1$ and $k$ in such a way that the sum of the labels along any edge is no
greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees,
and they are counted by a generalised version of the Catalan numbers. We prove
a surprisingly simple refined counting formula, where we count trees with a
prescribed number of labels of each kind. Several corollaries are derived from
this formula, and an analogous theorem is proven for $k$-noncrossing trees, a
similarly defined family of labelled noncrossing trees that are related to
$(2k+1)$-ary trees.
