Updates on a cache-oblivious B-tree
Perhaps one must consider the parent of the node to be inserted as having weight 2. Then the "find the lowest in-density ancestor" is equivalent to "find the least amount of rebuilding needed to maintain the invariants".

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In the physical world, density is defined as mass divided by volume. This corresponds to the density notion in these cache-oblivious B-trees if we view each "item" as having mass 1 and each location in $\mathcal{T&#39;}$ (the static binary tree) as having volume 1.

Before inserting an item, we want to make sure that once it is in the tree it is in a subtree that is within its prescribed density bounds. We therefore imagine that it is already there in the tree, but taking up no space yet. The reason why we might imagine that it presently takes up no space is that, well, there might not be any space for it to take up right now, as in your example.

To make sure that the tree the new item will be in is within bounds, we need to find its closest ancestor that we actually can rebalance (without increasing the height) and insert the item into while still remaining within the density boundaries. To do so, we find the nearest ancestor that will be within the density boundaries, which means using its current "volume" and its soon-to-be-correct "mass". To get the latter, we must include an extra unit of mass for the item we are about to insert.