  In this paper, a natural bijection between multichains of binary paths and
shifted tableaux is presented, and it is used for the enumeration of the chains
with maximum length from a given path $P$ to the maximum path
$\mathbf{1}_{|P|}$. By mapping chains to shifted tableaux, the main formulas
given in a recent paper by the authors for the enumeration of the $P -
\mathbf{1}_{|P|}$ chains having only small intervals and minimum length are
proved, using some new bijections on shifted tableaux.
