Self-similar Markov trees form a remarkable family of random compact real trees equipped with a decoration function that is positive on the skeleton, encompassing the Brownian continuum random tree, stable Lévy trees, fragmentation trees, and growth-fragmentation trees. In this book, the authors develop a consistent and unified theory of self-similar Markov trees by bringing together and vastly generalizing results that had been scattered across the random tree literature over several decades. They begin with in-depth coverage of the construction of self-similar Markov trees, then address the study of self-similar Markov trees in the continuous. In Part II, the authors build on this material and introduce readers to current research. They establish general invariance principles for Galton–Watson trees with integer types and illustrate them through numerous combinatorial classes of random trees that have appeared in the literature.

‘This book provides a detailed study of the class of self-similar Markov trees, which encompasses most of the classical models of continuous random trees. Scaling limits of discrete models are thoroughly discussed, as well as connections with random planar geometry. This book will be a must to anybody interested in continuous genealogical structures.’

Jean-François Le Gall - Université Paris-Saclay

‘This wonderful book gives an account of some of the latest developments in the theory of random trees and their scaling limits. It has been written with great care and clarity and is illustrated throughout with beautiful simulations that really bring the mathematics to life. An essential reference for the field.’

Christina Goldschmidt - University of Oxford

‘Bertoin, Curien, and Riera elegantly guide the reader through this unified treatment of self-similar Markov trees. Enriched with examples and motivated by scaling limits appearing in random planar maps and combinatorial random trees, the reader enjoys insights throughout drawn from Lévy processes, self-similar Markov processes, real trees, and branching processes.'

Andreas Kyprianou - University of Warwick

'By introducing the powerful concept of Self-Similar Markov trees, Bertoin, Curien, and Riera provide a unifying viewpoint on random structures of diverse origins. This superbly written book will therefore be highly relevant to graduate students and researchers, be they interested in classical objects of probability theory, like branching processes and continuum random trees, which are given here a rejuvenating cure, or in modern developments of random geometry and statistical physics, like random planar maps and conformally invariant random structures.'

Grégory Miermont - École normale supérieure de Lyon