"abound, swarm, be prolific," Old English teman (Mercian), tieman (West Saxon) "beget, give birth to, bring forth, produce, propagate," from Proto-Germanic *tau(h)mjan (denominative), from PIE root *deuk- "to lead." Related to team (n.) in its now-obsolete Old English sense of "family, brood of young animals." The meaning "abound, swarm" is first recorded 1590s, on the notion of "be full of as if ready to give birth." Related: Teemed; teeming.
"to flow copiously," early 14c., "to empty out" (transitive), from a Scandinavian source akin to Old Norse toema "to empty," from tomr "empty," cognate with Old English tom (adj.) "empty, free from." The original notion is of "to empty a vessel," thus "to pour out." Intransitive sense of "to pour, flow, stream" is from 1828. Related: Teemed; teeming.