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We show that the definition of fractality cannot be applied to networks with \u2018tree-like\u2019 structure and exponential growth rate of neighbourhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs, we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komj\u00e1thy and Simon, Song\u2013Havlin\u2013Makse model, spherically symmetric trees and supercritical Galton\u2013Watson trees.<\/jats:p>","DOI":"10.1093\/comnet\/cnz005","type":"journal-article","created":{"date-parts":[[2019,1,23]],"date-time":"2019-01-23T17:42:21Z","timestamp":1548265341000},"page":"764-791","source":"Crossref","is-referenced-by-count":6,"title":["Transfinite fractal dimension of trees and hierarchical scale-free graphs"],"prefix":"10.1093","volume":"7","author":[{"given":"J\u00falia","family":"Komj\u00e1thy","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, MB Eindhoven, Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roland","family":"Molontay","sequence":"additional","affiliation":[{"name":"Department of Stochastics, Budapest University of Technology and Economics, 1521 Budapest, Hungary and MTA-BME Stochastics Research Group, 1521 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K\u00e1roly","family":"Simon","sequence":"additional","affiliation":[{"name":"Department of Stochastics, Budapest University of Technology and Economics, 1521 Budapest, Hungary and MTA-BME Stochastics Research Group, 1521 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2019,2,23]]},"reference":[{"key":"2019101609185783600_B1","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/PL00012580","article-title":"Connected components in random graphs with given expected degree sequences","volume":"6","author":"Chung,","year":"2002","journal-title":"Ann. 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