\contentsline {section}{\numberline {1}Introduction}{2}
\contentsline {subsection}{\numberline {1.1}Correspondence with results of \cite {jp01vol}}{4}
\contentsline {section}{\numberline {2}Random mappings and the forest volume formula}{4}
\contentsline {paragraph}{Random Mappings.}{5}
\contentsline {paragraph}{Rooted forests.}{6}
\contentsline {paragraph}{The forest volume formula.}{7}
\contentsline {section}{\numberline {3}Random Forests}{8}
\contentsline {paragraph}{Constructions of $p$-trees.}{9}
\contentsline {paragraph}{Coalescent construction of $p$-forests.}{10}
\contentsline {subsection}{\numberline {3.1}Distribution of the roots of a $p$-forest}{10}
\contentsline {subsection}{\numberline {3.2}Conditioning on the set of roots.}{11}
\contentsline {paragraph}{Conditioning a $p$-mapping to have no cycles within a given set}{12}
\contentsline {paragraph}{Conditioning a $p$-mapping on its set of cyclic points}{12}
\contentsline {paragraph}{Descriptions of a $p$-forest with roots $R$.}{13}
\contentsline {paragraph}{Distribution of level sets.}{15}
\contentsline {section}{\numberline {4}Spanning Subtrees}{15}
\contentsline {subsection}{\numberline {4.1}Joyal's bijection between marked rooted trees and mappings}{20}
\contentsline {section}{\numberline {5}Hurwitz distributions}{21}
\contentsline {subsection}{\numberline {5.1}Constructions from $p$-mappings}{22}
\contentsline {subsection}{\numberline {5.2}Constructions from $p$-trees}{24}
\contentsline {subsection}{\numberline {5.3}Distribution of tree components}{25}
\contentsline {subsection}{\numberline {5.4}A Hurwitz multinomial distribution}{27}
\contentsline {subsection}{\numberline {5.5}Percolation probabilities}{28}
\contentsline {paragraph}{Unriented percolation.}{28}
\contentsline {paragraph}{Oriented percolation.}{29}
\contentsline {section}{\numberline {6}Restrictions of $p$-forests}{30}
\contentsline {subsection}{\numberline {6.1}Proof of the restriction theorem}{32}
\contentsline {paragraph}{Completion of the proof.}{34}
\contentsline {subsection}{\numberline {6.2}Variations}{34}
\contentsline {subsection}{\numberline {6.3}Thinning of $p$-forests}{37}