\BOOKMARK [1][-]{section.1}{Introduction}{}% 1
\BOOKMARK [2][-]{subsection.1.1}{Composition of the paper}{section.1}% 2
\BOOKMARK [1][-]{section.2}{The random trees}{}% 3
\BOOKMARK [2][-]{subsection.2.1}{m-ary search trees}{section.2}% 4
\BOOKMARK [2][-]{subsection.2.2}{General preferential attachment trees}{section.2}% 5
\BOOKMARK [1][-]{section.3}{Main results}{}% 6
\BOOKMARK [2][-]{subsection.3.1}{Fringe subtrees in random m-ary search trees}{section.3}% 7
\BOOKMARK [2][-]{subsection.3.2}{Protected nodes in random m-ary search trees}{section.3}% 8
\BOOKMARK [2][-]{subsection.3.3}{Fringe subtrees in preferential attachment trees}{section.3}% 9
\BOOKMARK [1][-]{section.4}{Generalised P\363lya urns}{}% 10
\BOOKMARK [1][-]{section.5}{P\363lya urns to count fringe subtrees in random m-ary search trees}{}% 11
\BOOKMARK [1][-]{section.6}{Proofs of Theorem 3.2 and Theorem 3.5}{}% 12
\BOOKMARK [1][-]{section.7}{P\363lya urns to count fringe subtrees in preferential attachment trees}{}% 13
\BOOKMARK [2][-]{subsection.7.1}{A P\363lya urn with infinitely many types for the general case}{section.7}% 14
\BOOKMARK [2][-]{subsection.7.2}{A P\363lya urn with finitely many types for the linear case}{section.7}% 15
\BOOKMARK [1][-]{section.8}{Proofs of Theorem 3.9 and Theorem 3.11}{}% 16
\BOOKMARK [1][-]{section.9}{A P\363lya urn to count protected nodes in m-ary search trees}{}% 17
\BOOKMARK [1][-]{section.10}{Proof of Theorem 3.7}{}% 18
\BOOKMARK [1][-]{section.11}{Examples with explicit calculations of variances}{}% 19
\BOOKMARK [2][-]{subsection.11.1}{Example of Theorem 3.5 when m=3 and k=4}{section.11}% 20
\BOOKMARK [2][-]{subsection.11.2}{Example of Theorem 3.2 when m=4}{section.11}% 21
\BOOKMARK [2][-]{subsection.11.3}{Example of Theorem 3.11 when k=3}{section.11}% 22
\BOOKMARK [1][-]{section.12}{Degrees}{}% 23
\BOOKMARK [2][-]{subsection.12.1}{Out-degrees in m-ary search trees}{section.12}% 24
\BOOKMARK [2][-]{subsection.12.2}{Out-degrees in preferential attachment trees}{section.12}% 25
\BOOKMARK [1][-]{appendix.A}{The covariance matrix \040in Section 11.2}{}% 26
\BOOKMARK [1][-]{appendix.B}{The covariances in the matrix \040in Section 11.3}{}% 27
