We explore a sample of 148 solar-like stars to search for a possible correlation between the slopes of the abundance trends versus condensation temperature (known as the Tc slope) with stellar parameters and Galactic orbital parameters in order to understand the nature of the peculiar chemical signatures of these stars and the possible connection with planet formation. We find that the Tc slope significantly correlates (at more than 4σ) with the stellar age and the stellar surface gravity. We also find tentative evidence that the Tc slope correlates with the mean galactocentric distance of the stars (Rmean), suggesting that those stars that originated in the inner Galaxy have fewer refractory elements relative to the volatiles. While the average Tc slope for planet-hosting solar analogs is steeper than that of their counterparts without planets, this difference probably reflects the difference in their age and Rmean. We conclude that the age and probably the Galactic birth place are determinant to establish the star's chemical properties. Old stars (and stars with inner disk origin) have a lower refractory-to-volatile ratio. Based on observations collected with the HARPS spectrograph at the 3.6-m telescope (072.C-0488(E)), installed at the La Silla Observatory, ESO (Chile), with the UVES spectrograph at the 8-m Very Large Telescope program IDs: 67.C-0206(A), 074.C-0134(A), 075.D-0453(A), installed at the Cerro Paranal Observatory, ESO (Chile), and with the UES spectrograph at the 4.2-m William Herschel Telescope, installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, on the island of La Palma.Appendix A is available in electronic form at http://www.aanda.org