Continued Fractions and Catalan Problems
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.