Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow s=\frac{\log(m)}{-\log\varepsilon}$. Is $s$ the "local dimension" of the tree?
Is there a relationship between the the Hausdorff dimension of a Cantor tree and its local dimension?
The notation I'm using is similar to that found in Falconer's "Fractal Geometry":
By the "$k$th level of the Cantor tree", I am referring to whatever remains of the original interval after $k-1$ steps of removing intervals.
If I have a tree for which the number of children at each level is constant, say $M_k$ is the number of children of each interval in level $k-1$ and $D_k$ is the supremum of all the intervals in the $kth$ level, what does the upper bound for the Hausdorff dimension given by $\frac{\log(M_1...M_{k-1})}{-\log(D_k)}$ represent? I think it is the local dimension as well, but I am not sure.