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In this paper, we introduce the Push Tree problem which contains elements from both the Steiner Tree and the Shortest Path problem. The Push Tree problem deals with the trade-offs between the push and pull mechanisms used in information distribution and retrieval. We present some initial complexity results and analyse several heuristics. Moreover, we discuss what lessons can be learned from the static and deterministic Push Tree problem for more realistic scenarios characterised by high uncertainty and changing information request and update patterns.