We propose TrendSegment, a methodology for detecting multiple change-points corresponding to linear trend changes or point anomalies in one-dimensional data. A core ingredient of TrendSegment is a new Tail-Greedy Unbalanced Wavelet transform: a conditionally orthonormal, bottom-up transformation of the data through an adaptively constructed unbalanced wavelet basis, which results in a sparse representation of the data. The bottom-up nature of this multiscale decomposition enables the detection of point anomalies or linear trend changes at once as the decomposition focuses on local features in its early stages and on global features next. To reduce the computational complexity, the proposed method merges multiple regions in a single pass over the data. We show the consistency of the estimated number and locations of change-points. The practicality of our approach is demonstrated through simulations and two real data examples, involving Iceland temperature data and sea ice extent of the Arctic and the Antarctic.