The name "Calennig" comes from the Latin "Kalends," which was the name for a Roman New Year's festival. The Romans gave each other olive branches as a way of wishing others good health for the new year.
In Wales, the offering is traditionally an apple, stuck with cloves, with an evergreen branch (usually boxwood, but sometimes pine or holly) through its stem and three small sticks of wood to form a tripod to support the apple. This "perllan" would be taken door to door; in the "olden times" the children begged for food, but in more modern times the children would receive some coins or candy.
A town called Cwm Gwaun and other villages in the surrounding area still celebrate this old custom on January 13, the first day of the new year on the old Julian Calendar.
Read more about the Welsh new year traditions on Hen Galan (Welsh for "first day of the month").