Signals on hierarchical trees can be viewed as a generalization of discrete signals of length 2^N. In this work, we extend the classic discrete Haar wavelets to a Haar-like wavelet basis that works for signals on hierarchical trees. We first construct a specific wavelet basis and give its inverse and normalized transform matrices.
As analogue to the classic case, operators and wavelet generating functions are constructed for the tree structure. This leads to the definition of multiresolution analysis on a hierarchical tree. We prove the previously selected wavelet basis is an orthogonal multiresolution. Classification of all possible wavelet basis that generate an orthogonal multiresolution is then given. In attempt to find more efficient encoding and decoding algorithms, we construct a second wavelet basis and show that it is also an orthogonal multiresolution. The encoding and decoding algorithms are given and their time complexity are analyzed.
In order to link change of tree structure and encoded signal, we define weighted hierarchical tree, tree cut and extension. It is then shown that a simply relation can be established without the need for global change of the transform matrix. Finally, we apply thresholding to the transform and give an upper bound of error.