We consider expansions which give arbitrary orthonormal tilings of the time-frequency plane. These differ from the short-time Fourier transform, wavelet transform, and wavelet packets tilings in that they change over time. We show how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. One method is based on lapped orthogonal transforms, which makes it possible to change the number of channels in the transform. A second method is based on the construction of orthogonal boundary filters; these allow us to construct essentially arbitrary tilings. We present a double-tree algorithm which for a given signal decides on the best binary segmentation, and which tree split to use for each segment. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g. rate-distortion). This gives best time-varying bases. Results of experiments on test signals are shown.