The problem of learning queries from tree structured data is studied by this paper. A tree structured data is modeled as a node-labeled tree T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document}, and applying a query q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} on T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document} will return a set q(T)\documentclass[12pt]{minimal}
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\begin{document}$$q(T)$$\end{document} which is a subset of nodes in T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document}. For a tree-node pair (T,t)\documentclass[12pt]{minimal}
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\begin{document}$$(T,t)$$\end{document} where t\documentclass[12pt]{minimal}
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\begin{document}$$t$$\end{document} is a node in T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document}, q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} is called to accept the pair if t∈q(T)\documentclass[12pt]{minimal}
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\begin{document}$$t\in {q(T)}$$\end{document}, and reject the pair if t∉q(T)\documentclass[12pt]{minimal}
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\begin{document}$$t\notin {q(T)}$$\end{document}. For some query class L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{L }$$\end{document}, given tree-node pair sets Ep\documentclass[12pt]{minimal}
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\begin{document}$$E_p$$\end{document} and En\documentclass[12pt]{minimal}
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\begin{document}$$E_n$$\end{document}, the tree query learning problem is to find a query q∈L\documentclass[12pt]{minimal}
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\begin{document}$$q\in \mathcal{L }$$\end{document} such that (1) q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} rejects all pairs in En\documentclass[12pt]{minimal}
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\begin{document}$$E_n$$\end{document}, and (2) the size of pairs in Ep\documentclass[12pt]{minimal}
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\begin{document}$$E_p$$\end{document} accepted by q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} is maximized. On four different query classes Q/\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /}$$\end{document}, Q/,∗\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,*}$$\end{document}, Q/,//\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,//}$$\end{document} and Q/,[]\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,[]}$$\end{document}, this paper studies the hardness of the corresponding tree query learning problems. For Q/\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /}$$\end{document}, a PTime algorithm is given. For Q/,∗\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,*}$$\end{document} and Q/,//\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,//}$$\end{document}, the NP-complete results are shown. For Q/,[]\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,[]}$$\end{document}, the problem is shown to be NP-hard by considering two constrained fragments of Q/,[]\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,[]}$$\end{document}. Also, for Q/,∗\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,*}$$\end{document}, Q/,[]\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,[]}$$\end{document} and Q/,//\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal Q ^{\tiny /,//}$$\end{document}, it is shown that there are no n1−ϵ\documentclass[12pt]{minimal}
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\begin{document}$$n^{1-\epsilon }$$\end{document}-approximation algorithms for any ϵ>0\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon >0$$\end{document}.
