The pilot reported that he was departing an asphalt surfaced runway in his tundra tire and tailwheel-equipped airplane in crosswind conditions. During the departure roll, and just as the airplane began to lift off the airplane yawed to the left. In an effort to correct for they yaw, he applied right rudder. The right main tire contacted the runway's surface and the airplane veered to the right. He applied left rudder and brake to correct for the veer to no avail. The airplane continued a slow right turn exited the runway and impacted trees sustaining substantial damage to the left wing struts. The pilot reported no preaccident mechanical failures or malfunctions that would have precluded normal operations. In the recommendation section of the NTSB Accident/Incident Reporting Form 6120.1, the pilot stated that the accident may have been prevented if he had utilized a gravel surfaced runway.