Mythical tree problem

The mythical tree problem is a combinatorial problem created by Harvey Friedman.

In the problem, the tree has a strange growth pattern. In the first stage, the tree grows k branches, forming k treetops. In the next stage, one of the treetops forms k+1 branches, increasing the number of treetops. The problem asks: what is the maximum number of branch segments a tree starting at k branches can have, provided that a squirrel can go from the root to any treetop without navigating more than four branch segments?

Friedman has shown that:


 * k = 2 corresponds to 21,990,232,555,518 segments
 * k = 3 corresponds to a value > 22 95
 * k = 4 corresponds to a value > 2 ↑↑ 22 95
 * The growth rate of the number of segments as a function of k is similar to that of the Ackermann function.