B+tree height after delete: PostgreSQL fast root

Many databases use B+tree indexes, but they all differ. It's a sorted structure. The leaf pages are logically sorted so that a specific key value belongs to one page. A lookup by value reaches a single leaf page and either directly finds an entry for that value or immediately knows there's no entry with that key. When a page becomes full, it is split into two pages, each covering its own dedicated range. To find the right page, an internal page holds the range of values for the pages below. This internal page can become full, and a new level is added above it. Finally, at the highest level, there's a single internal page that is the root. A lookup always starts at the root and goes down to the leaves, following the branches of internal pages. In a traditional B+tree lookup, the cost is proportional to the height of the tree because the search starts at the root and descends to a leaf: 1 page to read when all fits in one leaf that is also the root (0 levels of internal pages, total height is 1). With small keys, this level can typically index hundreds of rows. 2 pages to read when there's one root that can list all leaf pages (1 level of internal page, total height is 2). With small keys, this level can typically index tens or hundreds of thousands of rows. 3 pages to read when there's one level of branches under the root (so 2 levels of internal pages, total height is 3). With small keys, this level can typically index millions of rows. This means that finding one key within ten million rows may require traversing 3 index pages, where most of them are probably in cache given the small number of branches compared to the leaves. For a given index size, whatever the value you are looking for, it's always the same number of pages to read because the index is balanced (the commonly accepted meaning of the B in B+tree). This property is maintained because any page can split, but only splitting the root adds another level. I've described how the height of an index can incr