ash-tray (n.)

"powdery remains of fire," Middle English asshe, from Old English æsce "ash," from Proto-Germanic *askon (source also of Old Norse and Swedish aska, Old High German asca, German asche, Middle Dutch asche, Gothic azgo "ashes"), from PIE root *as- "to burn, glow." Spanish and Portuguese ascua "red-hot coal" are Germanic loan-words.

An ancient symbol of grief or repentance; hence Ash Wednesday (c. 1300), from the custom introduced by Pope Gregory the Great of sprinkling ashes on the heads of penitents on the first day of Lent. Ashes meaning "mortal remains of a person" is attested from late 13c., in reference to the ancient custom of cremation. The meaning "finely pulverized lava thrown from a volcano" is from 1660s.

Old English treg, trig "flat wooden board with a low rim," from Proto-Germanic *traujam (source also of Old Swedish tro, a corn measure), from PIE *drou-, variant of root *deru- "be firm, solid, steadfast," with specialized senses "wood, tree" and derivatives referring to objects made of wood. The primary sense may have been "wooden vessel."