  We use a continued fraction approach to compare two statistical ensembles of
quadrangulations with a boundary, both controlled by two parameters. In the
first ensemble, the quadrangulations are bicolored and the parameters control
their numbers of vertices of both colors. In the second ensemble, the
parameters control instead the number of vertices which are local maxima for
the distance to a given vertex, and the number of those which are not. Both
ensembles may be described either by their (bivariate) generating functions at
fixed boundary length or, after some standard slice decomposition, by their
(bivariate) slice generating functions. We first show that the fixed boundary
length generating functions are in fact equal for the two ensembles. We then
show that the slice generating functions, although different for the two
ensembles, simply correspond to two different ways of encoding the same
quantity as a continued fraction. This property is used to obtain explicit
expressions for the slice generating functions in a constructive way.
