This is a very extensive and thorough reference book on enumerating trees with analytical methods. Trees are of quintessential importance in several parts of mathematics, such as Combinatorics, Probability, and the Theory of Algorithms, so it is the reviewer's task to tell which of these branches is the best served by any book. For this book, the answer is Probability. More results are formulated in probabilistic terms than in any other terms.
There is a wealth of information concerning the distribution of various parameters of random trees, such as their height, or insertion depth, or number of leaves. The asymptotics of the limiting distributions are also discussed.
This is the kind of book that people will open to get results they need. After looking up the results, the readers will close the book, because it is largely a collection of results; it does not contain much in terms of motivation, context, or discussion. Prospective readers are encouraged to consult the detailed, six-page table of contents that is available on at the Springer web site (linked to in our table of contents page). That will do a better service to the book than a reviewer's report can.
It is unlikely that the book would be used as a textbook for several reasons. First, the topic may be a little bit too specialized. Second, there is the issue of continuity. For instance, Generating Functions are defined and treated in Chapter 2, but they are used before that, in Chapter 1. Third, students, even graduate students, are likely to need more motivation and context than what is provided here. Finally, there are no exercises in the book, though there are many examples.
Miklós Bóna is Associate Professor of Mathematics at the University of Florida.