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The recently introduced twin-width of a graph G is the minimum integer d such that G has a d-contraction sequence, that is, a sequence of |V (G)| − 1 iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most d, where a red edge appears between two sets of identified vertices if they are not homogeneous in G (not fully adjacent nor fully non-adjacent). We show that if a graph admits a d-contraction sequence, then it also has a linear-arity tree of f (d)-contractions, for some function f . Informally if we accept to worsen the twinwidth bound, we can choose the next contraction from a set of Θ(|V (G)|) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most n!c n graphs labeled by [n], for some constant c. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an O(log n)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture.We then explore the small conjecture that, con- * This work was supported by the grants from French National Agency under PRC program (project Digraphs, ANR-19-CE48-0013-01), under JCJC program (project ASSK, ANR-18-CE40-0025-01), and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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