Local fractal dimension and wavelet analysis to investigate patterns of trees in insulators

Trees in insulators have been investigated using fractal geometry. The complexity of the tree is estimated by only one number of the fractal dimension. In some cases that the local shape of the branches of the trees is different in another branches, it is not useful to analyze whole tree. Tree should be multifractal. Therefore multifractal analysis is done and then a global spectrum is obtained. It is, however, difficult to understand the curve of the global spectrum (Global dimension and singularity) because both numbers mean dimension. We sometimes need a local information on the tree. Multifractal, however, does not tell us it because the local information is lost when it is calculated. We suggested new numerical method for trees; local fractal dimension. Fractal dimension at each point on the tree is calculated. The correlation function is used for estimating local fractal dimension. We furthermore suggest wavelet analysis instead of correlation function. For the correlation function, the area calculated is determined by someone. In the case of wavelet function, the area is, however, automatically determined, because the wavelet function changes their supported region automatically. It is easy to point out the change of the tree and it is useful to estimate the development of the tree using these methods.