We present the novel adaptive hierarchical sensing algorithm K-AHS, which samples sparse or compressible signals with a measurement complexity equal to that of Compressed Sensing (CS). In contrast to CS, K-AHS is adaptive as sensing vectors are selected while sampling, depending on previous measurements. Prior to sampling, the user chooses a transform domain in which the signal of interest is sparse. The corresponding transform determines the collection of sensing vectors. K-AHS gradually refines initial coarse measurements to significant signal coefficients in the sparse transform domain based on a sensing tree which provides a natural hierarchy of sensing vectors. K-AHS directly provides significant signal coefficients in the sparse transform domain and does not require a reconstruction stage based on inverse optimization. Therefore, the K-AHS sensing vectors must not satisfy any incoherence or restricted isometry property. A mathematical analysis proves the sampling complexity of K-AHS as well as a general and sufficient condition for sampling the optimal k-term approximation, which is applied to particular signal models. The analytical findings are supported by simulations with synthetic signals and real world images. On standard benchmark images, K-AHS achieves lower reconstruction errors than CS.