New Notation
We define an (i,d)-forest, where 0 <= i <= d, as an i-dimensionally (partially)
ordered forest of (i+1)-dimensionally (partially) ordered forests of ... of
(d-1)-dimensionally (partially) ordered forests of d-dimensional trees.
A node with no i-dimensional children for all i < j can be interpreted as the
root of a (j,d)-forest. The forest construction operation then becomes
X(r_(d-1), r_(d-2), ..., r_0), where r_i represents an (i,d)-forest. Note that
this means that an (j,d)-forest can only be constructed if the (i,d)-forests
used represent empty trees for all i < j. By these definitions, if r_i is the
empty tree for all i < j, then the newly constructed node labeled X can be
interpreted as a (j,d)-forest. If the forest so constructed can be interpreted
as a (d,d)-forest, then its root has no successors apart from the d dimension,
and so it represents a d-dimensional tree, which fits well with our definition
of a (d,d)-forest.
Posted by lemanal at July 2, 2004 05:44 PM