A tsunami eqrthquake is defined as a shock which generates extensive tsunamis but relatively weak seismic waves. A comparative study is made for the two recent tsunami earthquakes, and a subduction mechanism near a deep-sea trench is discussed. These two earthquakes occurred at extremely shallow depths far off the coasts of the Kurile Islands and of eastern Hokkaido on October 20, 1963, and on June 10, 1975, respectively. Both can be regarded as an aftershock of the preceding larger events. Their tsunami heights and seismic wave amplitudes are compared with those of the preceding events. The results show that the time constants involved in the tsunami earthquakes are relatively long but not long enough to explain the observed disproportionality between the tsunamis and the seismic waves. The process times are estimated to be less than 100 s. The spatio-temporal characteristics of the two events suggest that they represent a seaward and upward extension of the rupture associated with a great earthquake which did not break the free surface at the coseismic stage. The amplitude and phase spectra of long-period surface waves and the long-period P waveforms indicate that this extension of the rupture did not take place entirely along the lithospheric interface emerging as a trench axis. It rather branched upward from the interface in a complex way through the wedge portion at the leading edge of the continental lithosphere. This wedge portion consists in large part of thick deformable sediments. A large vertical deformation and hence extensive tsunamis result from such a branching process. A shallowest source depth, steepening of rupture surfaces, and a deformable nature of the source region all enhance generation of tsunamis. The wedge portion ruptured by a tsunami earthquake is usually characterized by a very low seismic activity which is presumably due to ductility of the sediments. We suggest that this portion fractures in a brittle way to generate a tsunami earthquake when it is loaded suddenly by the occurrence of a great earthquake and that otherwise it yields slowly. Upward branching of the rupture from the lithospheric interface produces permanent deformation of the free surface which is relative uplift landward and relative subsidence trenchward of the zone of surface break. This surface break zone geomorphologically corresponds to the lower continental slope between the deep-sea terrace and the trench. Such a mode of permanent deformation seems to be consistent with a rising feature of the outer ridge of the deep-sea terrace and a depressional feature of the trench. This consistency implies a causal relationship betwen great earthquake activities and geomorphological features near the trench.