  We give upper and lower bounds for the Hausdorff dimensions for a class of
graph-directed measures when its underlying directed graph is the infinite
N-ary tree. These measures are different from graph-directed self-similar
measures driven by finite directed graphs and are not necessarily Gibbs
measures. However our class contains several measures appearing in fractal
geometry and functional equations, specifically, measures defined by
restrictions of non-constant harmonic functions on the two-dimensional
Sierp\'inski gasket, the Kusuoka energy measures on it, and, measures defined
by solutions of de Rham's functional equations driven by linear fractional
transformations.
