\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{OEIS}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{sec:intro}{{1}{1}{Introduction}{section.1}{}}
\citation{Hinz}
\citation{Hinz}
\citation{HanoiTower}
\citation{CircSet}
\citation{OEIS}
\citation{A000975}
\citation{EC2}
\citation{CatMod}
\citation{Lord}
\newlabel{eq:An}{{1}{2}{Introduction}{equation.1.1}{}}
\citation{CatMod}
\@writefile{toc}{\contentsline {section}{\numberline {2}Leaf depths in binary trees}{3}{section.2}}
\newlabel{sec:prelim}{{2}{3}{Leaf depths in binary trees}{section.2}{}}
\newlabel{prop:wedge}{{1}{4}{}{theorem.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A binary operation on binary trees}}{4}{figure.1}}
\newlabel{fig:wedge}{{1}{4}{A binary operation on binary trees}{figure.1}{}}
\newlabel{example:correspondence}{{3}{4}{}{theorem.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Correspondence between binary trees and parenthesizations}}{5}{figure.2}}
\newlabel{fig:T2P}{{2}{5}{Correspondence between binary trees and parenthesizations}{figure.2}{}}
\newlabel{lem:DNE}{{4}{5}{}{theorem.4}{}}
\newlabel{lem:alternating}{{5}{5}{}{theorem.5}{}}
\newlabel{lem:admissible}{{6}{6}{}{theorem.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Main results}{6}{section.3}}
\newlabel{sec:main}{{3}{6}{Main results}{section.3}{}}
\newlabel{eq:refine}{{2}{6}{Main results}{equation.3.2}{}}
\newlabel{prop:sign}{{7}{7}{}{theorem.7}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces $C_{\ominus ,n,r}$ for $0\le n\le 12$ and $0\le r\le n+1$}}{7}{table.1}}
\newlabel{tab:Cnr}{{1}{7}{$C_{\ominus ,n,r}$ for $0\le n\le 12$ and $0\le r\le n+1$}{table.1}{}}
\newlabel{thm:Cnr}{{8}{7}{}{theorem.8}{}}
\newlabel{thm:ominus}{{9}{8}{}{theorem.9}{}}
\newlabel{eq:skipping}{{3}{8}{Main results}{equation.3.3}{}}
\newlabel{prop:result}{{10}{9}{}{theorem.10}{}}
\bibcite{CatMod}{1}
\bibcite{Hinz}{2}
\bibcite{HanoiTower}{3}
\bibcite{CircSet}{4}
\bibcite{Lord}{5}
\bibcite{OEIS}{6}
\@writefile{toc}{\contentsline {section}{\numberline {4}Questions}{10}{section.4}}
\newlabel{sec:questions}{{4}{10}{Questions}{section.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Acknowledgments}{10}{section.5}}
\bibcite{EC2}{7}
\bibcite{A000975}{8}