  We present a further analysis of the fragmentation at heights of the
normalized Brownian excursion. Specifically we study a representation for the
mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable
subordinator and use it to study its jumps; this accounts for a description of
how a typical fragment falls apart. These results carry over to the height
fragmentation of the stable tree. Additionally, the sizes of the fragments in
the Brownian fragmentation when it is about to reduce to dust are described in
a limit theorem.
