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By combining calls into a tree structure, we allow each sub tree to run in parallel reducing the contention we normally see on a single parent node (left hand picture). This is still a safe barrier since the root of the tree is the only node that will notify the children of a lock release.


Comparing traffic and space to the previous slide, this takes $O(P)$ space and, where $P=2^k$, $2\sum_{i=0}^{k-1} 2^i = 2P-2$ traffic (including the release).

Note that in the picture above, on the right, there are double the amount of arrows we actually need (because for each parent, one of the children should be electrically equivalent to the parent). Then, we must account for the release.