  The Brownian continuum tree was extensively studied in the 90s as a universal
random metric space. One construction obtains the continuum tree by a change of
metric from an excursion function (or continuous circle mapping) on $[0,1]$.
This change of metric can be applied to all excursion functions, and generally
to continuous circle mappings. In 2008, Picard proved that the dimension theory
of the tree is connected to its associated contour function: the upper box
dimension of the continuum tree coincides with the variation index of the
contour function.
  In this article we give a short and direct proof of Picard's theorem through
the study of packings. We develop related and equivalent notions of variations
and variation indices and study their basic properties. Finally, we link the
dimension theory of the tree with the dimension theory of the graph of its
contour function.
