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Constructing Small Tree Grammars and Small Circuits for Formulas Hucke, Danny Lohrey, Markus Noeth, Eric grammar-based compression tree compression arithmetical circuits It is shown that every tree of size n over a fixed set of sigma different ranked symbols can be decomposed into O(n/log_sigma(n)) = O((n * log(sigma))/ log(n)) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O(n/log_sigma(n)), which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of O(n/log^{0.19}(n)) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m <= n different variables occur, can be transformed (in time O(n * log(n)) into an arithmetical circuit of size O((n * log(m))/log(n)) and depth O(log(n)). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Danny Hucke and Markus Lohrey and Eric Noeth 2014 Is Part Of LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2014.457 urn:nbn:de:0030-drops-48639 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.457 eng https://creativecommons.org/licenses/by/3.0/legalcode