Graph theory and multiset intersections

Hi

I'm currently pondering the problem of how to count the number of
isomorphic graphs (well trees actually) that you will obtain if you
intersect bags of the nodes in the tree. Each bag basically contains
all the nodes in a given sub-tree. A further question I'm trying to
answer is how many bags are needed so that you can uniquely reconstruct
a tree from the intersection of each of the bags. Does anyone have any
pointers to papers or textbooks, or google keywords that might help me
find an answer to these problems?

Hiren

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