A similar entry appears in the Wheel of Time Companion confirming the information available in the main story arc.
Avendoraldera was a cutting from the Tree of Life, Avendesora taken by the Aiel and presented to the Cairhienin in return for their assistance during the Breaking, when the ancestors of the Cairhienin were the only people who allowed the Aiel free access to water (TSR, Ch. 34). They called the Cairheinin the Watersharers and allowed them passage through the waste, insisting that their trains of merchant wagons fly a banner bearing the trefoil leaf of Avendesora. For four hundred years (TSR, Ch. 39), this brought riches of silk and ivory to Cairhien (TDR, Ch. 39).
Tigraine's disappearance in 972NE led to a series of plots culminating in Laman's decision to chop down Avendoraldera to make a new throne. The Aiel response to this was to launch an attack, something they saw as an execution, but was viewed as an invasion by the other nations who joined together to fight them, leading to the Aiel War (TSR, Ch. 34).