When graph database users explore unfamiliar graphs, potentially with heterogeneous structure, users may need to find how two or more groups of nodes are connected in a graph, even when users are not able to describe the connections. This is only partially supported by existing query languages, which allow searching for paths, but not for trees connecting three or more node groups. In this work, we formally show how to integrate connecting tree patterns (CTPs, in short) with a graph query language such as GPML, SPARQL or Cypher, leading to Extended Queries (or EQs, in short). We then study a set of algorithms for evaluating CTPs; we generalize prior keyword search work to be complete, most importantly by (i) considering bidirectional edge traversal, (ii) allowing users to select any score function for ranking CTP results and (iii) returning all results. To cope with very large search spaces, we propose efficient pruning techniques and formally establish a large set of cases where our best algorithm, MOLESP, is complete even with pruning. Our experiments validate the performance of our algorithms on many synthetic and real-world workloads.