  Birman, Ko and Lee have introduced a new monoid ${\cal B}^{*}_{n}$--with an
explicit presentation--whose group of fractions is the $n$-strand braid group
${\cal B}_{n}$. Building on a new approach by Digne, Michel and himself, Bessis
has defined a {\it dual} braid monoid for every finite Coxeter type Artin-Tits
group extending the type A case. Here, we give an explicit presentation for
this dual braid monoid in the case of types B and D, and we study the
combinatorics of the underlying Garside structures.
