What is the literature on a sparse matrix encoding of rose trees?
In view of rose tree being just a special case of a graph,
the matrix representation you mentioned is simply the adjacency matrix representation of directed graph. Your particular encoding takes the rose tree's edge to always be directed consistently either from parent to child or child to parent.

Questions about rose tree in this encoding thus can be researched easily by looking up results about adjacency matrix representation of directed tree graph.

Relevant notions are:


get all leaves of subtree -> reachability
graft tree -> wedge sum