Chordal graph and its clique tree
Chordal graphs can be defined as intersection graph of subtrees of any tree.  So the answer to your decision question is trivially YES.  On the construction side, for each subtree $T_v$ of bags (it's convenient and conventional to call the nodes of the tree as bags), you'll have a unique new vertex $v$, which is put into all bags of the $T_v$.