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This book grew out of a common passion for a beautiful natural object that wedecidedtocallthe valuativetree .Motivatedbyquestionsstemmingfrom complex dynamics and complex analysis, we realized that we needed to - derstand the link between valuations, which are purely algebraic objects, and more geometric or analytic constructions such as blowups or Lelong numbers. More precisely, we looked at the structure of a special set of valuations, and wefoundthatthisspacehadaveryrichanddelicatetopologicalstructure.We hope that the reader will share our enthusiasm while progressively exploring this space into its ?ner details all along this book. This monograph has bene?ted from the help of many people. The ?rst author wishes to warmly thank Bernard Teissier for his constant support and help, and Patrick Popescu-Pampu,Mark Spivakovsky and Michel Vaqui e for fruitful discussions. The second author expresses his gratitude to Jean- Francois Lafont, Robert Lazarsfeld and Karen Smith. We both thank the referees for a number of useful suggestions. Our work was done at several places, including the Institut de Math e- tiques in Paris,the Department of Mathematics of the University of Michigan in Ann Arbor, RIMS in Kyoto and IMPA in Rio de Janeiro. We are grateful to theseinstitutions for havingprovidedus anexcellentand motivating at- sphere for working on our project. During the writing period, the ?rst author was supported by the CNRS, whereas the second author was supported by STINT and by NSF Grant No DMS-0200614.
