The distribution of height and diameter in random non-plane binary trees

Authors:
Nicolas Broutin;Philippe Flajolet
Affiliations:
Algorithms Project, INRIA-Rocquencourt, Le Chesnay F-78153, France;Algorithms Project, INRIA-Rocquencourt, Le Chesnay F-78153, France
Venue:
Random Structures & Algorithms
Year:
2012

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Abstract

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees (“Otter trees”), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size n is proved to admit a limiting theta distribution, both in a central and local sense, and obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 (Supported by French ANR Project BOOLE; ANR-09-BLAN-0011.)