A family of subsets has a tree on the same vertex set making each member a subtree exactly when it has the Helly property, a chordal intersection graph, and a well-foundedness condition that rules out certain infinite descending chains.
The intersection graphs of subtrees in trees are exactly the chordal graphs
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The vertex sets of subtrees of a tree
A family of subsets has a tree on the same vertex set making each member a subtree exactly when it has the Helly property, a chordal intersection graph, and a well-foundedness condition that rules out certain infinite descending chains.